Unit 9: Thermodynamics
Weightage : 15–18%
A resistor of resistance \(R\) is sealed in a closed container with \(n\) moles of gas inside. A battery of emf \(\varepsilon\) is connected to the resistor. Which of the following graphs shows the correct relationship between the gas atoms’ average velocity \((v_{\mathrm{avg}})\) and the electrical energy \((E)\) supplied to the resistor?
Correct Answer: \( \boxed{\mathrm{C}} \)
The electrical energy supplied to the resistor is converted into thermal energy, which increases the internal energy and temperature of the gas.
The average translational kinetic energy of a gas particle is
\(K_{\mathrm{avg}}=\frac{1}{2}mv_{\mathrm{avg}}^{2}.\)
Assuming the electrical energy supplied to the resistor is transferred to the gas,
\(E\propto K_{\mathrm{avg}}\propto v_{\mathrm{avg}}^{2}.\)
Therefore,
\(v_{\mathrm{avg}}\propto\sqrt{E}.\)
A square-root relationship increases rapidly at first and then gradually levels off (it is increasing with decreasing slope), which corresponds to graph (C).
Hence, the correct graph is \(\boxed{\mathrm{C}}\).
An ideal gas with molecules of mass \(m\) is contained in a cube with sides of area \(A\). The average vertical component of the velocity of the gas molecules is \(v\), and \(N\) molecules hit the side of the cube in a time \(\Delta t\). What is the pressure exerted by the gas on the bottom of the cube?
Correct Answer: \( \boxed{\mathrm{D}} \)
Pressure is defined as force per unit area:
\( P=\dfrac{F}{A} \)
Each gas molecule striking the bottom surface reverses its vertical momentum, producing a change in momentum of
\( \Delta p = 2mv \).
Since \(N\) molecules collide with the surface during the time interval \(\Delta t\), the total rate of change of momentum (force) is
\( F=\dfrac{2Nmv}{\Delta t} \).
Therefore, the pressure on the bottom surface is
\( P=\dfrac{F}{A}=\dfrac{2Nmv}{A\Delta t} \).
Thus, the correct answer is (D).
Two samples of ideal gas in separate containers have the same number of molecules and the same temperature, but the molecular mass of gas \(X\) is greater than that of gas \(Y\). Which of the following correctly compares the average speed of the molecules of the gases and the average force the gases exert on their respective containers?
| Option | Average Speed of Molecules | Average Force on Container |
|---|---|---|
| (A) | Greater for gas \(X\) | Greater for gas \(X\) |
| (B) | Greater for gas \(X\) | The forces cannot be compared without knowing the volumes of the gases. |
| (C) | Greater for gas \(Y\) | Greater for gas \(Y\) |
| (D) | Greater for gas \(Y\) | The forces cannot be compared without knowing the volumes of the gases. |
Correct Answer: \( \boxed{\mathrm{D}} \)
For an ideal gas, the average molecular kinetic energy depends only on the absolute temperature:
\( \dfrac{1}{2}m\overline{v^2}=\dfrac{3}{2}k_{\mathrm{B}}T. \)
Therefore,
\( v_{\mathrm{rms}}=\sqrt{\dfrac{3k_{\mathrm{B}}T}{m}}. \)
Since gas \(X\) has the larger molecular mass, its molecules move more slowly. Thus, the average molecular speed is greater for gas \(Y\).
The average force exerted on a container depends on the gas pressure,
\( P=\dfrac{Nk_{\mathrm{B}}T}{V}. \)
Although the gases have the same number of molecules and the same temperature, the container volumes are not given. Without knowing the volumes (or equivalently the pressures or wall areas), the average forces exerted on the containers cannot be compared.
Therefore, the correct answer is (D).
