Thermodynamics

State functions are properties of a system that depend only on the current state of the system, not on the path used to get to that state. They are intrinsic properties of the system and include properties like pressure, temperature, volume, internal energy, enthalpy, entropy, and Gibbs free energy.

Option A: Internal Energy

Internal energy, denoted as $U$, is a state function. It is the total energy contained within the system, including kinetic and potential energy at the molecular level. Internal energy changes in a system only depend on the initial and final states of the system, regardless of the process or path taken to achieve these states. Therefore, it fits the criteria of a state function.

Option B: Irreversible Expansion Work

Irreversible expansion work is not a state function. Work, in general, is a path function because it depends on the path taken during a process. Whether the path involves irreversible or reversible processes, work includes energy transfer that depends significantly on how that transfer is carried out. Therefore, irreversible expansion work depends on the specific details of the process and is not solely determined by the initial and final states of the system.

Option C: Reversible Expansion Work

Similarly to irreversible expansion work, reversible expansion work is also a path function and not a state function. Even though reversible processes are ideal and involve quasi-static changes that maintain the system in near equilibrium throughout, the work done (or reversible expansion work) during such processes still depends on the specific path taken. This includes how slowly the process is carried out and the intermediate steps, distinguishing it from a state function.

Option D: Molar Enthalpy

Molar enthalpy, denoted as $H$, is a state function. It is defined as the sum of the internal energy $U$ of a system plus the product of the pressure $P$ and volume $V$ of the system, multiplied by the number of moles $n$, expressed as:

$H = U + PV$

Since $U$, $P$, and $V$ are all state functions, their combination into enthalpy continues to be dependent solely on the state of the system, not on the path taken to reach that state.

Conclusion:

Among the given options, Internal energy (Option A) and Molar enthalpy (Option D) are state functions, while both Irreversible expansion work (Option B) and Reversible expansion work (Option C) are not state functions. They are path functions.

We have

C(graphite) $\to$ C(diamond)

For solid at constant temperature, Gibbs energy can be calculated as

$dG = \Delta Vdp$

$\int\limits_{\Delta {G_1}}^{\Delta {G_2}} {d(\Delta {G_T}) = \int\limits_{{p_1}}^{{p_2}} {\Delta Vdp} } $

$\Delta {G_2} - \Delta {G_1} = \Delta V({p_2} - {p_1})$

$(2.9 \times {10^3} - 0) = 2 \times {10^{ - 6}}({p_2} - 1)$

$({p_2} - 1) = {{2.9 \times {{10}^3}} \over {2 \times {{10}^{ - 6}}}}$ Pa

$({p_2} - 1) = {{2.9 \times {{10}^3}} \over {2 \times {{10}^{ - 6}} \times {{10}^5}}}$ bar

p₂ = 14501 bar

Work done in an irreversible isothermal process is given by :

$ \begin{aligned} \mathrm{W}_{\mathrm{irr}} & =-\mathrm{P}_{\mathrm{ext}}\left(\mathrm{V}_2-\mathrm{V}_1\right) \\\\ & =-3.0 \mathrm{~atm}(2.0 \mathrm{~L}-1.0 \mathrm{~L}) \\\\ & =-3.0 \times 1.0 \mathrm{~atm} \mathrm{~L} \\\\ \mathrm{~W}_{\mathrm{irr}} & =-3.0 \mathrm{~L} \mathrm{~atm} \\\\ \mathrm{~W}_{\mathrm{irr}} & =-3.0 \times 101.3 \mathrm{~J} \,\,\,\,\,\,\,\,\,\ [1 \mathrm{~L} \mathrm{~atm}=101.3] \\\\ & =-303.9 \mathrm{~J} \end{aligned} $

At a constant pressure,

$ \begin{aligned} & \Delta \mathrm{U}=\mathrm{q}_{\mathrm{p}}+\mathrm{W}_{\text {irr }}=\mathrm{q}_{\mathrm{p}}-\mathrm{P}_{\text {ext }}\left(\mathrm{V}_2-\mathrm{V}_1\right) \\\\ & \Delta \mathrm{U}=\Delta \mathrm{H}-\mathrm{P}_{\text {ext }}\left(\mathrm{V}_2-\mathrm{V}_1\right) \\\\ & \Delta \mathrm{H}=\Delta \mathrm{U}+\mathrm{P}_{\text {ext }}\left(\mathrm{V}_2-\mathrm{V}_1\right) \end{aligned} $

Since, reaction takes place at constant temperature; hence, $\Delta \mathrm{U}=0$.

$ \begin{aligned} \Delta \mathrm{H} & =\mathrm{P}_{\text {ext }}\left(\mathrm{V}_2-\mathrm{V}_1\right)=-\mathrm{W}_{\mathrm{irr}}=303.9 \mathrm{~J} \\\\ \Delta \mathrm{H}_{\mathrm{sys}} & =303.9 \mathrm{~J} \\\\ \Delta \mathrm{S}_{\text {sys }} & =\frac{\Delta \mathrm{H}_{\text {sys }}}{\mathrm{T}} \text { and } \\\\ \end{aligned} $

Finally, the change in entropy of the surroundings (ΔS_surr) is equal to the negative of the change in entropy of the system, because the system and its surroundings form an isolated system :

ΔS_surr = -ΔS_sys

$ \begin{aligned} \Delta \mathrm{S}_{\text {surr }} & =-\frac{\Delta \mathrm{H}_{\text {sys }}}{\mathrm{T}}=-\frac{303.9 \mathrm{~J}}{300 \mathrm{~K}} \\\\ & =-1.013 \mathrm{~J} \mathrm{~K}^{-1} \end{aligned} $

(A) $\to$ (R and T)

Freezing of water,

H₂O_(l) $\mathrel{\mathop{\kern0pt\rightleftharpoons} \limits_{273\,K}^{1\,atm}} $ H₂O_(s)

The system is cooled i.e.; heat is released during the process

so, q < 0

$\mathop {Water}\limits_{(Less\,volume)} $ $\rightleftharpoons$ $\mathop {Ice}\limits_{(More\,volume)} $ + heat

Volume is increased i.e.; $\Delta$V = +ve.

w = $-$P$\Delta$V = $-$ve

i.e.; w < 0 (expansion)

Entropy of system is decreased, $\Delta$S_sys < 0.

$\Delta$U = q + w

As q < 0, w < 0 so, $\Delta$U < 0.

At equilibrium, $\Delta$G = 0.

(B) $\to$ (P, Q and S)

Expansion of 1 mol of an ideal gas into a vacuum under isolated conditions,

w = 0, q = 0 so, $\Delta$U = 0

For expansion, $\Delta$S_sys > 0 as entropy increases.

$\Delta G = - nRT\ln {{{V_2}} \over {{V_1}}}$

For expansion, V₂ > V₁

$\Delta$G = $-$ve i.e.; $\Delta$G < 0.

(C) $\to$ (P, Q and S)

Mixing of equal volumes of two ideal gases at constant temperature and pressure in an isolated container.

q = 0 (isolated)

w = $-$P$\Delta$V

w = 0 ($\because$ $\Delta$V = 0)

$\Delta$S_sys > 0 (mixing of gases)

$\Delta$U = q + w = 0

$\Delta$G = $\Delta$H $-$ T$\Delta$S

$\Delta$G = q_p $-$ T$\Delta$S (at constant P, T)

$\Delta$G = 0 $-$ T$\Delta$S = $-$T$\Delta$S

$\Delta$G < 0 ($\because$ $\Delta$S_sys > 0)

(D) $\to$ (P, Q, S and T)

$\mathop {{H_{2(g)}}}\limits_{(300\,K)} \mathrel{\mathop{\kern0pt\rightleftharpoons} \limits_{Cool}^{Heat,\,1\,atm}} \mathop {{H_{2(g)}}}\limits_{(600\,K)} $

Internal energy (U), entropy (S) and free energy (G) are state functions which depend only upon the state of the system and do not depend upon the path by which the state is attained.

Thus, $\Delta$U = 0, $\Delta$S = 0 and $\Delta$G = 0

Work and heat are path functions but the same path is retraced so, q = 0 and w = 0.

To analyze the options provided, let's discuss the changes in entropy for the system and the surroundings separately during the phase change from liquid water (H₂O(l)) to water vapor (H₂O(g)) at 100°C and 1 atmosphere pressure. This is a typical boiling process.

First, let's consider the change in entropy, $\Delta S_{\text{system}}$, of the water itself (the system). When water boils and changes from a liquid to a gas, the molecules spread out significantly, occupying a larger volume and showing greater randomness in their motion. This transition from a more ordered phase (liquid) to a less ordered phase (gas) results in an increase in entropy. Thus, $\Delta S_{\text{system}}>0$.

Next, let's consider the change in entropy of the surroundings, $\Delta S_{\text{surroundings}}$. When the water boils at 100°C, it must absorb heat to do so. This heat is typically taken from the surroundings. As the surroundings lose heat, according to the formula for entropy change $\Delta S = \frac{q}{T}$, where $q$ is the heat transferred and $T$ is the temperature, the surroundings lose entropy if the process is happening at a constant temperature as typically assumed in such cases. Therefore, here $\Delta S_{\text{surroundings}}<0$ as heat loss by surroundings (negative $q$) results in a decrease in entropy.

Combining these insights, we find:

• The entropy of the system increases ($\Delta S_{\text{system}}>0$).
• The entropy of the surroundings decreases ($\Delta S_{\text{surroundings}}<0$).

Therefore, the correct answer is: Option B: $\Delta S_{\text{system}}>0$ and $\Delta S_{\text{surroundings}}<0$.

From the graph, as the term pV increases, the temperature increases; and when pV decreases the temperature decreases.

The term PV increases K$\to$L, so this involves heating and is an isobaric process.

It increases from N$\to$K, so this involves heating and is an isochoric process.

It decreases from M$\to$N, so this involves cooling and is an isobaric process.

It decreases from L$\to$M, so this involves cooling and is an isochoric process.

From the above solution, we have

From K$\to$L : heating (isobaric)

From L$\to$M : cooling (isochoric)

From M$\to$N : cooling (isobaric)

From N$\to$K : heating (isochoric)

To find the standard enthalpy of combustion of glucose, we will first write the balanced chemical equation for the combustion of glucose. The chemical formula for glucose is $ C_6H_{12}O_6 $. Combustion involves the reaction of a substance with oxygen to produce carbon dioxide (CO₂) and water (H₂O) typically. The reaction for the combustion of glucose can be represented as:

$ C_6H_{12}O_6 (s) + 6O_2 (g) \rightarrow 6CO_2 (g) + 6H_2O (l) $

Each molecule of glucose (C6H12O6) produces six molecules of CO₂ and six molecules of H₂O. Now, we use the given standard enthalpies of formation (ΔH_f°) for these substances to find the enthalpy of the reaction. The enthalpy of a reaction can be calculated using the formula:

$ \Delta H_{\text{reaction}} = \sum \Delta H_{f,\text{products}} - \sum \Delta H_{f,\text{reactants}} $

For the given reaction:

• ΔH_f° of CO₂(g) = –400 kJ/mol
• ΔH_f° of H₂O(l) = –300 kJ/mol
• ΔH_f° of glucose(s) = –1300 kJ/mol

The sum of the standard enthalpies of formation of the products:

$ [6 \times (-400) \text{ kJ/mol}] + [6 \times (-300) \text{ kJ/mol}] = (-2400 + -1800) \text{ kJ} = -4200 \text{ kJ} $

The sum of the standard enthalpies of formation of the reactants (glucose as only part contributing with enthalpy value other than zero):

$ [-1300 \text{ kJ/mol}] + [6 \times 0 \text{ kJ/mol for } O_2] $

Therefore, the standard enthalpy of combustion of glucose is:

$ \Delta H_{\text{reaction}} = (-4200 \text{ kJ}) - (-1300 \text{ kJ}) = -2900 \text{ kJ} $

So, -2900 kJ of energy is released per mole of glucose combusted at standard conditions. The standard enthalpy of combustion per gram of glucose can be calculated using the molar mass of glucose. Glucose has a molar mass of approximately 180 g/mol.

$ \Delta H_{\text{combustion per gram}} = \frac{-2900 \text{ kJ/mol}}{180 \text{ g/mol}} = -16.11 \text{ kJ/g} $

Thus, the correct answer is -16.11 kJ per gram of glucose, which corresponds to option C –16.11 kJ.

We know that

Binding energy = Total energy of reactants $-$ Total energy of products

So, we have

225 = 2(C $-$ C) + 1(H $-$ H) $-$ 2(C $-$ H) + 1(C $\equiv$ C)

225 = 1410 + 330 $-$ 2 $\times$ 350 + (C $\equiv$ C)

Solving, we get the value of C $\equiv$ C as 815 kJ mol^$-$1.

Analysis of each transformation:

(A) CO₂(s) $\to$ CO₂(g)

This transformation represents the sublimation of solid carbon dioxide (dry ice) into gaseous carbon dioxide. This process is:

Phase transition (p): The substance changes from solid to gas phase.

$\Delta H$ is positive (r): Sublimation is an endothermic process, meaning it absorbs heat from the surroundings.

$\Delta S$ is positive (s): The entropy increases as the substance moves from a more ordered solid state to a less ordered gaseous state.

(B) CaCO₃(s) $\to$ CaO(s) + CO₂(g)

This transformation is the thermal decomposition of calcium carbonate to form calcium oxide and carbon dioxide. This process involves:

$\Delta H$ is positive (r): Decomposition reactions are typically endothermic, requiring energy to break chemical bonds.

$\Delta S$ is positive (s): The process results in an increase in entropy, primarily because a gas (CO₂) is produced from a solid compound.

(C) 2H $\to$ H₂(g)

This is the formation of hydrogen gas (H₂) from atomic hydrogen (H). This transformation:

$\Delta S$ is negative (t): Two individual hydrogen atoms combine into a diatomic molecule, decreasing the system's entropy since fewer particles are free to move independently.

(D) P_{(white, solid)} $\to$ P_{(red, solid)}

Crystalline solid white phosphorous is converted to amorphous red phosphorous. This process involves a phase change. Since, white and red phosphorous are allotropic form of phosphorous, i.e., both are composed of phosphorous atoms but the connectivity between atom differ, it is therefore regarded as allotropic change.

The white form of phosphorous has lot of angular strain as compared to stable red phosphorous. Hence, randomness of system due to angular strain decreases as red phosphorus is formed. The entropy change is negative $(\Delta S<0)$

Option Evaluation:

Upon reviewing each transformation and comparing the mappings:

Option A agrees completely with the above analysis with proper mapping for each statement.

Thus, Option A is correct:

A $\to$ p,r,s; B $\to$ r,s; C $\to$ t; D $\to$ p,q,t

The bond energy of a C-C (carbon-carbon) single bond is a measure of the strength of that bond, or how much energy is required to break one mole of such bonds in a gaseous substance. This value is a fundamental concept in chemistry, particularly in the study of organic molecules and reactions.

For a C-C single bond, the bond energy is typically around 83 kcal/mol (kcal per mole). This places the value closest to option C amongst the given choices.

Therefore, the correct answer is:

Option C: 100

The standard molar enthalpy of formation, denoted as $\Delta H_f^\circ$, is defined as the change in enthalpy when one mole of a compound is formed from its constituent elements in their standard states under standard conditions (298 K and 1 atm pressure). By definition, the standard molar enthalpy of formation of a pure element in its most stable form at 298 K is zero.

Looking at the options:

• Br₂ (g) - Bromine's standard state at room temperature is liquid (Br₂(l)), not gas.
• Cl₂ (g) - Chlorine's standard state at room temperature is a gas, which means the standard molar enthalpy of formation for Cl₂(g) is indeed zero.
• H₂O (g) - Water in gaseous form is not a basic element but a compound of hydrogen and oxygen, so its enthalpy of formation is not zero.
• CH₄ (g) - Methane, represented here, is also a compound composed of carbon and hydrogen, thus its enthalpy of formation is also not zero.

Therefore, the correct option is Option B, Cl₂ (g), as chlorine gas is a diatomic molecule and an element in its standard state at 298 K, and as such, its standard molar enthalpy of formation is zero.

In reality, work can be converted to heat entirely but not vice versa. This is in accordance with second law of thermodynamics. So, no process is possible in practice where heat is entirely converted to work.

Take for example the heat engine. It has to reject some heat to the sink and convert the remainder into work. Thus, entire heat supplied is not converted into work. On the other hand, in a refrigerator, work has to be supplied to transfer heat from a colder body to hotter surroundings-in this case, the entire work input is converted to heat.

In practice, reversible isothermal processes are impossible because heat transfer always takes place across a finite temperature difference. This is an irreversibility. Thus, the entropy of the system will always increase if there is a heat transfer to the system. In that case, the amount of heat supplied to the system cannot entirely be converted to work (unlike reversible isothermal processes where heat is entirely converted to work).

So, both the statements are clearly true but the second one does not explain the first one.

Process :

$ \text{H}_2\text{O}(l,\,1\,\text{bar},\,373\,\text{K}) \to \text{H}_2\text{O}(g,\,1\,\text{bar},\,373\,\text{K}) $

Step 1. Identify what temperature and pressure mean

At 373 K and 1 bar, this is the normal boiling point of water (approximately, at 1 atm ≈ 1.013 bar).

So, at this condition, liquid water and water vapor are in equilibrium.

Step 2. Thermodynamic definitions

At equilibrium between two phases at a given $T$ and $p$:

$ \Delta G = 0 $

because the two phases have equal molar Gibbs energies.

So for a reversible phase change at equilibrium:

$ \boxed{\Delta G = 0} $

Step 3. Entropy change

When liquid water vaporizes,

• Disorder increases,

• Number of microstates increases,

so

$ \boxed{\Delta S > 0} $

(positive entropy change).

✅ Therefore, the correct set is:

$ \boxed{\Delta G = 0, \ \Delta S = +\text{ve}} $

✅ Correct option: A

$(\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ )$

$ = - 54.07 \times 1000 - 298 \times 10$ J/mol

$ = - 54070 - 2980 = 57050$

$\Delta G^\circ = - 2.303\,RT\,{\log _{10}}\,K$

$ - 57050 = - 2.303 \times 298 \times 8.314\,{\log _{10}}K$

$\log K = 10$

Assume the pressure of a monoatomic gas is P and volume is V at the temperature T .

As given, $\frac{\mathrm{P}}{\mathrm{V}}=1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)$

From equation (i), it can be concluded that,

$ P=V \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$

According to the first law of thermodynamics,

$ d q=\mathrm{C}_v d \mathrm{~T}+p d \mathrm{~V}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(iii)$

Here, $d q$ stands for the differential increase in the amount of added heat in the system, $\mathrm{C}_v$ is the molar heat capacity at the constant volume, $d \mathrm{~T}$ is a small temperature difference, $p$ is the pressure and $d \mathrm{~V}$ represents the small change in volume.

For one mole of an ideal gas,

PV = RT

The above relation can be written as,

$ \mathrm{P} d \mathrm{~V}+\mathrm{V} d p=\mathrm{R} d \mathrm{~T}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(iv) $

Also, from equation (i),

$ \mathrm{P} d \mathrm{~V}=\mathrm{V} d p \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(v)$

Substituting in equation (iv), we get,

$ 2 \mathrm{P} d \mathrm{~V}=\mathrm{R} d \mathrm{~T} $

On rearranging,

$ \mathrm{P} d \mathrm{~V}=\frac{\mathrm{R}}{2} d \mathrm{~T} $

Substitute this value of $\mathrm{P} d \mathrm{~V}$ in equation (iii),

$ \begin{aligned} d q & =\mathrm{C}_v d \mathrm{~T}+\frac{\mathrm{R}}{2} d \mathrm{~T} \\ & =\left(\mathrm{C}_v+\frac{\mathrm{R}}{2}\right) d \mathrm{~T} \end{aligned} $

On differentiating and rearranging,

$ \int \frac{d q}{d \mathrm{~T}}=\mathrm{C}_v+\frac{\mathrm{R}}{2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(vi)$

For a monoatomic gas,

$ C_v=\frac{3}{2} R $

Substitute the value of $\mathrm{C}_v$ in equation (vi),

$ \begin{aligned} & & \int \frac{d q}{d \mathrm{~T}} & =\frac{3}{2} \mathrm{R}+\frac{\mathrm{R}}{2} \\ \therefore & & \int \frac{d q}{d \mathrm{~T}} & =\frac{4}{2} \mathrm{R} \end{aligned} $

Carried by give path as shown in the diagram. Hence, by addition of $\Delta S$ for $A$ to $C, C$ to $D$ and $D$ to $B$, the value of $\Delta S_{(A \rightarrow B)}$ can be obtained.

$ \begin{aligned} & \Delta \mathrm{S}_{(\mathrm{A} \rightarrow \mathrm{~B})}=\Delta \mathrm{S}_{(\mathrm{A} \rightarrow \mathrm{C})}+\Delta \mathrm{S}_{(\mathrm{C} \rightarrow \mathrm{D})}+\Delta \mathrm{S}_{(\mathrm{D} \rightarrow \mathrm{~B})} \\ & \therefore \Delta \mathrm{S}_{(\mathrm{A} \rightarrow \mathrm{~B})}=\Delta \mathrm{S}_{(\mathrm{A} \rightarrow \mathrm{C})}+\Delta \mathrm{S}_{(\mathrm{C} \rightarrow \mathrm{D})}-\Delta \mathrm{S}_{(\mathrm{B} \rightarrow \mathrm{D})} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)\end{aligned} $

As given,

$ \begin{aligned} \Delta \mathrm{S}_{(\mathrm{A} \rightarrow \mathrm{C})} & =50 \text { e.u. } \\ \Delta \mathrm{S}_{(\mathrm{C} \rightarrow \mathrm{D})} & =30 \text { e.u. } \\ \Delta \mathrm{S}_{(\mathrm{B} \rightarrow \mathrm{D})} & =20 \text { e.u. } \end{aligned} $

substituting the values in equation (i),

$ \begin{aligned} \therefore \quad \Delta \mathrm{S}_{(\mathrm{A} \rightarrow \mathrm{~B})} & =50+30-20 \\ \therefore \quad \Delta \mathrm{~S}_{(\mathrm{A} \rightarrow \mathrm{~B})} & =60 \text { e.u. } \end{aligned} $