Heat and Thermodynamics
The temperature of an ideal gas is increased from $200 \mathrm{~K}$ to $800 \mathrm{~K}$. If r.m.s. speed of gas at $200 \mathrm{~K}$ is $v_{0}$. Then, r.m.s. speed of the gas at $800 \mathrm{~K}$ will be:
A body cools in 7 minutes from $60^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$. The temperature of the surrounding is $10^{\circ} \mathrm{C}$. The temperature of the body after the next 7 minutes will be:
The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:
A source supplies heat to a system at the rate of $1000 \mathrm{~W}$. If the system performs work at a rate of $200 \mathrm{~W}$. The rate at which internal energy of the system increases is
The number of air molecules per cm$^3$ increased from $3\times10^{19}$ to $12\times10^{19}$. The ratio of collision frequency of air molecules before and after the increase in number respectively is:
A Carnot engine operating between two reservoirs has efficiency $\frac{1}{3}$. When the temperature of cold reservoir raised by x, its efficiency decreases to $\frac{1}{6}$. The value of x, if the temperature of hot reservoir is $99^\circ$C, will be :
For three low density gases A, B, C pressure versus temperature graphs are plotted while keeping them at constant volume, as shown in the figure.
The temperature corresponding to the point '$\mathrm{K}$' is :
A sample of gas at temperature $T$ is adiabatically expanded to double its volume. The work done by the gas in the process is $\left(\mathrm{given}, \gamma=\frac{3}{2}\right)$ :
$\left(P+\frac{a}{V^{2}}\right)(V-b)=R T$ represents the equation of state of some gases. Where $P$ is the pressure, $V$ is the volume, $T$ is the temperature and $a, b, R$ are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^{2}}{a}$, will be:
The average kinetic energy of a molecule of the gas is
The pressure of a gas changes linearly with volume from $\mathrm{A}$ to $\mathrm{B}$ as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be
The correct relation between $\gamma = {{{c_p}} \over {{c_v}}}$ and temperature T is :
Assertion A: Efficiency of a reversible heat engine will be highest at $-273^{\circ} \mathrm{C}$ temperature of cold reservoir.
Reason R: The efficiency of Carnot's engine depends not only on the temperature of the cold reservoir but it depends on the temperature of the hot reservoir too and is given as $\eta=\left(1-\frac{T_{2}}{T_{1}}\right)$
In the light of the above statements, choose the correct answer from the options given below
The pressure $(\mathrm{P})$ and temperature ($\mathrm{T})$ relationship of an ideal gas obeys the equation $\mathrm{PT}^{2}=$ constant. The volume expansion coefficient of the gas will be :
Heat is given to an ideal gas in an isothermal process.
A. Internal energy of the gas will decrease.
B. Internal energy of the gas will increase.
C. Internal energy of the gas will not change.
D. The gas will do positive work.
E. The gas will do negative work.
Choose the correct answer from the options given below :
Heat energy of 184 kJ is given to ice of mass 600 g at $-12^\circ \mathrm{C}$. Specific heat of ice is $\mathrm{2222.3~J~kg^{-1^\circ}~C^{-1}}$ and latent heat of ice in 336 $\mathrm{kJ/kg^{-1}}$
A. Final temperature of system will be 0$^\circ$C.
B. Final temperature of the system will be greater than 0$^\circ$C.
C. The final system will have a mixture of ice and water in the ratio of 5 : 1.
D. The final system will have a mixture of ice and water in the ratio of 1 : 5.
E. The final system will have water only.
Choose the correct answer from the options given below :
At 300 K, the rms speed of oxygen molecules is $\sqrt {{{\alpha + 5} \over \alpha }} $ times to that of its average speed in the gas. Then, the value of $\alpha$ will be
(used $\pi = {{22} \over 7}$)
Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: If $d Q$ and $d W$ represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics $d Q=d U-d W$.
Reason R: First law of thermodynamics is based on law of conservation of energy.
In the light of the above statements, choose the correct answer from the options given below:
Match List I with List II
| List I | List II | ||
|---|---|---|---|
| A. | Isothermal Process | I. | Work done by the gas decreases internal energy |
| B. | Adiabatic Process | II. | No change in internal energy |
| C. | Isochoric Process | III. | The heat absorbed goes partly to increase internal energy and partly to do work |
| D. | Isobaric Process | IV. | No work is done on or by the gas |
Choose the correct answer from the options given below :
The graph between two temperature scales P and Q is shown in the figure. Between upper fixed point and lower fixed point there are 150 equal divisions of scale P and 100 divisions on scale Q. The relationship for conversion between the two scales is given by :-
According to law of equipartition of energy the molar specific heat of a diatomic gas at constant volume where the molecule has one additional vibrational mode is :-
The root mean square velocity of molecules of gas is
A Carnot engine with efficiency 50% takes heat from a source at 600 K. In order to increase the efficiency to 70%, keeping the temperature of sink same, the new temperature of the source will be :
Let $\gamma_1$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $\gamma_2$ be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio, $\frac{\gamma_1}{\gamma_2}$ is :
In an Isothermal change, the change in pressure and volume of a gas can be represented for three different temperature; $\mathrm{T_3 > T_2 > T_1}$ as :
1 g of a liquid is converted to vapour at 3 $\times$ 10$^5$ Pa pressure. If 10% of the heat supplied is used for increasing the volume by 1600 cm$^3$ during this phase change, then the increase in internal energy in the process will be :
Given below are two statements :
Statement I : The temperature of a gas is $-73^\circ$C. When the gas is heated to $527^\circ$C, the root mean square speed of the molecules is doubled.
Statement II : The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules.
In the light of the above statements, choose the correct answer from the option given below :
Two plates $\mathrm{A}$ and $\mathrm{B}$ have thermal conductivities $84 ~\mathrm{Wm}^{-1} \mathrm{~K}^{-1}$ and $126 ~\mathrm{Wm}^{-1} \mathrm{~K}^{-1}$ respectively. They have same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of $\mathrm{A}$ and $\mathrm{B}$ are kept at $100^{\circ} \mathrm{C}$ and $0{ }^{\circ} \mathrm{C}$ respectively, then the temperature of the surface of contact in steady state is _____________ ${ }^{\circ} \mathrm{C}$.
Explanation:
We can use the formula for heat transfer rate through a plate:
$Q = kA\frac{T_2 - T_1}{d}$
Where Q is the heat transfer rate, k is the thermal conductivity, A is the surface area, T1 and T2 are the temperatures on either side of the plate, and d is the thickness of the plate.
For plate A:
$Q_A = k_A A\frac{T_A - T}{d}$
For plate B:
$Q_B = k_B A\frac{T - T_B}{d}$
Since the heat transfer rate is the same through both plates in steady state:
$Q_A = Q_B$
We can now substitute the given values for thermal conductivities and temperatures:
$84A\frac{100 - T}{d} = 126A\frac{T - 0}{d}$
Notice that the surface area (A) and thickness (d) are the same for both plates, so they cancel out:
$84(100 - T) = 126T$
Now, we can solve for T:
$8400 - 84T = 126T$
$210T = 8400$
$T = \frac{8400}{210}$
$T = 40$
So, the temperature of the surface of contact in steady state is $40{ }^{\circ} \mathrm{C}$.
A steel rod of length $1 \mathrm{~m}$ and cross sectional area $10^{-4} \mathrm{~m}^{2}$ is heated from $0^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$ without being allowed to extend or bend. The compressive tension produced in the rod is ___________ $\times 10^{4} \mathrm{~N}$. (Given Young's modulus of steel $=2 \times 10^{11} \mathrm{Nm}^{-2}$, coefficient of linear expansion $=10^{-5} \mathrm{~K}^{-1}$ )
Explanation:
The change in length of the rod when it is heated is given by the equation:
$\Delta L = L_0 \cdot \alpha \cdot \Delta T$
where
- $\Delta L$ is the change in length,
- $L_0$ is the original length,
- $\alpha$ is the coefficient of linear expansion, and
- $\Delta T$ is the change in temperature.
Substituting the given values:
$\Delta L = 1 \, \text{m} \cdot 10^{-5} \, \text{K}^{-1} \cdot 200 \, \text{K} = 0.002 \, \text{m}$
The rod is not allowed to extend or bend, so a stress is created in the rod. This stress can be calculated using Young's modulus (Y), which is the ratio of the stress (force per unit area, F/A) to the strain (change in length per unit length, $\Delta L / L_0$):
$Y = \frac{F/A}{\Delta L / L_0}$
Rearranging for F gives:
$F = Y \cdot A \cdot \frac{\Delta L}{L_0}$
Substituting the given values:
$F = 2 \times 10^{11} \, \text{N/m}^2 \cdot 10^{-4} \, \text{m}^2 \cdot \frac{0.002 \, \text{m}}{1 \, \text{m}} = 4 \times 10^{4} \, \text{N}$
So the compressive tension produced in the rod is $4 \times 10^{4} \, \text{N}$.
(Assume that the specific heat capacity of water remains constant over the temperature range of the water).
Explanation:
The amount of heat energy required to raise the temperature of a substance can be calculated as:
Q = m $ \times $ c $ \times $ ΔT
where Q is the heat energy required, m is the mass of the substance, c is its specific heat capacity, and ΔT is the change in temperature.
The time required to heat a substance can be calculated as :
t = ${Q \over P}$
where t is the time required, and P is the power of the heating device.
The actual power output of the heating device can be calculated as:
Pactual = Pinput $ \times $ efficiency
where P_input is the input power to the device and efficiency is the fraction of input power that is actually converted to useful power output.
Substituting the given values:
Q = 2 kg $ \times $ 4200 J/kg/K $ \times $ (60 - 10) = 2 kg $ \times $ 4200 J/kg/K $ \times $ 50 K = 4200 $ \times $ 50 $ \times $ 2 J = 420,000 J
Pinput = 2000 W = 2000 J/s
Pactual = 2000 $ \times $ 0.7 = 1400 J/s
t = ${Q \over {{P_{actual}}}}$ = ${{420,000} \over {1400}}$ J/s = 300 s
So, the time required to heat 2 kg of water from 10°C to 60°C is approximately 300 s.
Explanation:
Let the correct temperature be X$^\circ$C
$ \Rightarrow {{X - 0} \over {100 - 0}} = {{41 - 5} \over {95 - 5}} \Rightarrow X = 40$
$\Rightarrow$ Temperature is 273 + 40 K = 313 K
A hole is drilled in a metal sheet. At $\mathrm{27^\circ C}$, the diameter of hole is 5 cm. When the sheet is heated to $\mathrm{177^\circ C}$, the change in the diameter of hole is $\mathrm{d\times10^{-3}}$ cm. The value of d will be __________ if coefficient of linear expansion of the metal is $1.6\times10^{-5}/^\circ$C.
Explanation:
$ \begin{aligned} & =5 \times 1.6 \times 10^{-5}(177-27) \\\\ & =0.012 \mathrm{~cm} \\\\ & =12 \times 10^{-3} \mathrm{~cm} \\\\ & \text { so, } d=12 \end{aligned} $
A thermodynamic system is taken from an original state D to an intermediate state E by the linear process shown in the figure. Its volume is then reduced to the original volume from E to F by an isobaric process. The total work done by the gas from D to E to F will be
The root mean square speed of smoke particles of mass $5 \times 10^{-17} \mathrm{~kg}$ in their Brownian motion in air at NTP is approximately. [Given $\mathrm{k}=1.38 \times 10^{-23} \mathrm{JK}^{-1}$]
A vessel contains $14 \mathrm{~g}$ of nitrogen gas at a temperature of $27^{\circ} \mathrm{C}$. The amount of heat to be transferred to the gas to double the r.m.s speed of its molecules will be :
Take $\mathrm{R}=8.32 \mathrm{~J} \mathrm{~mol}^{-1} \,\mathrm{k}^{-1}$.
A Carnot engine has efficiency of $50 \%$. If the temperature of sink is reduced by $40^{\circ} \mathrm{C}$, its efficiency increases by $30 \%$. The temperature of the source will be:
Given below are two statements :
Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature.
Statement II : The rms speed of oxygen molecules in a gas is $v$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2 v$.
In the light of the above statements, choose the correct answer from the options given below :
In $1^{\text {st }}$ case, Carnot engine operates between temperatures $300 \mathrm{~K}$ and $100 \mathrm{~K}$. In $2^{\text {nd }}$ case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in $2^{\text {nd }}$ case) will be :
Which statements are correct about degrees of freedom ?
(A) A molecule with n degrees of freedom has n$^{2}$ different ways of storing energy.
(B) Each degree of freedom is associated with $\frac{1}{2}$ RT average energy per mole.
(C) A monatomic gas molecule has 1 rotational degree of freedom where as diatomic molecule has 2 rotational degrees of freedom.
(D) $\mathrm{CH}_{4}$ has a total of 6 degrees of freedom.
Choose the correct answer from the options given below :
If $K_{1}$ and $K_{2}$ are the thermal conductivities, $L_{1}$ and $L_{2}$ are the lengths and $A_{1}$ and $A_{2}$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_{2}}{K_{1}}=9, \frac{A_{1}}{A_{2}}=2, \frac{L_{1}}{L_{2}}=2$. Then, for the arrangement as shown in the figure, the value of temperature $\mathrm{T}$ of the steel - copper junction in the steady state will be:
Read the following statements :
A. When small temperature difference between a liquid and its surrounding is doubled, the rate of loss of heat of the liquid becomes twice.
B. Two bodies $P$ and $Q$ having equal surface areas are maintained at temperature $10^{\circ} \mathrm{C}$ and $20^{\circ} \mathrm{C}$. The thermal radiation emitted in a given time by $\mathrm{P}$ and $\mathrm{Q}$ are in the ratio $1: 1.15$.
C. A Carnot Engine working between $100 \mathrm{~K}$ and $400 \mathrm{~K}$ has an efficiency of $75 \%$.
D. When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice.
Choose the correct answer from the options given below :
Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $1: 4$, then
A. The r.m.s. velocity of gas molecules in two vessels will be the same.
B. The ratio of pressure in these vessels will be $1: 4$.
C. The ratio of pressure will be $1: 1$.
D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $1: 4$.
Choose the correct answer from the options given below :
An ice cube of dimensions $60 \mathrm{~cm} \times 50 \mathrm{~cm} \times 20 \mathrm{~cm}$ is placed in an insulation box of wall thickness $1 \mathrm{~cm}$. The box keeping the ice cube at $0^{\circ} \mathrm{C}$ of temperature is brought to a room of temperature $40^{\circ} \mathrm{C}$. The rate of melting of ice is approximately :
(Latent heat of fusion of ice is $3.4 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$ and thermal conducting of insulation wall is $0.05 \,\mathrm{Wm}^{-1 \circ} \mathrm{C}^{-1}$ )
A gas has $n$ degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be :
7 mol of a certain monoatomic ideal gas undergoes a temperature increase of $40 \mathrm{~K}$ at constant pressure. The increase in the internal energy of the gas in this process is :
(Given $\mathrm{R}=8.3 \,\mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )
A monoatomic gas at pressure $\mathrm{P}$ and volume $\mathrm{V}$ is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be :
Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is $\sqrt2$ times the speed of sound, then the value of n will be :