A black body is at a temperature of $500 \text{ K}$. It emits energy at a rate which is proportional to: (1997)
According to Stefan-Boltzmann law, the rate of emission of radiant energy by a black body is proportional to the fourth power of its absolute temperature, $E \propto T^4$. Here, $T = 500 \text{ K}$, so $E \propto (500)^4$.
If the temperature of the sun is doubled, the rate of energy received on earth will be increased by a factor of: (1998)
The rate of energy received on earth is directly proportional to the rate of energy radiated by the sun. By Stefan's law, $E \propto T^4$. If $T$ is doubled, $E' \propto (2T)^4 = 16 T^4 = 16E$.
The total radiant energy per unit area, normal to the direction of incidence, received at a distance $R$ from the center of a star of radius $r$, whose outer surface radiates as a black body at a temperature $T$ $K$ is given by: (2010 Pre)
(Where $\sigma$ is Stefan’s Constant)
Radiant energy per unit area per unit time (Intensity) at distance $R$ is $I = \frac{P}{4\pi R^2} = \frac{\sigma (4\pi r^2) T^4}{4\pi R^2} = \frac{\sigma r^2 T^4}{R^2}$.
Two cylinders A and B of equal capacity are connected to each other via a stop cock. A contains an ideal gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stop cock is suddenly opened. The process is: (2020)
As the entire system is thermally insulated, no heat exchange occurs with the surroundings ($Q = 0$). Therefore, the free expansion process is adiabatic.
In which of the following processes, heat is neither absorbed nor released by a system? (2019)
In an adiabatic process, the system is insulated from its surroundings, meaning the net heat exchange ($Q$) is zero.
A sample of $0.1 \text{ g}$ of water at $100^{\circ}\text{C}$ and normal pressure ($1.013 \times 10^{5} \text{ Nm}^{-2}$) requires $54 \text{ cal}$ of heat energy to convert to steam at $100^{\circ}\text{C}$. If the volume of the steam produced is $167.1 \text{ cc}$, the change in internal energy of the sample, is: (2018)
Heat supplied $Q = 54 \text{ cal} = 54 \times 4.18 \text{ J} = 225.72 \text{ J}$. Work done $W = P\Delta V = 1.013 \times 10^{5} \times (167.1 - 0.1) \times 10^{-6} \approx 16.92 \text{ J}$. By first law of thermodynamics, $\Delta U = Q - W = 225.72 - 16.92 = 208.8 \text{ J}$, which is closest to $208.7 \text{ J}$.
13. A monoatomic gas at a pressure $P$, having a volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. The final pressure of the gas is (take $\gamma = 5/3$): (2014)
For isothermal process $P_1V_1 = P_2V_2 \Rightarrow P \times V = P_2 \times 2V \Rightarrow P_2 = P/2$. For adiabatic process $P_2V_2^{\gamma} = P_3V_3^{\gamma} \Rightarrow (P/2)(2V)^{5/3} = P_3(16V)^{5/3} \Rightarrow P_3 = P/64$.
14. During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of $\frac{C_p}{C_v}$ for the gas is: (2013)
$P \propto T^3$ or $P T^{-3} = \text{constant}$. We know that for an adiabatic process $P^{1-\gamma} T^{\gamma} = \text{constant}$ or $P T^{\frac{\gamma}{1-\gamma}} = \text{constant}$. Comparing the powers of $T$, $\frac{\gamma}{1-\gamma} = -3 \Rightarrow \gamma = -3 + 3\gamma \Rightarrow 2\gamma = 3 \Rightarrow \gamma = 3/2$.
11. An ideal gas is compressed to half its initial volume by means of several processes. Which of the process results in the maximum work done on the gas? (2015 Re)
Work done on the gas is maximum in the adiabatic process as the area under the $P-V$ curve is maximum.
In thermodynamic processes which of the following statements is not true? (2009)
An isochoric process is one in which volume remains constant, not pressure. Therefore, statement (a) is incorrect.