Question 151:
moderateA 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 $E \propto (500)^4$.
Question 152:
If the radius of a star is $R$ and it acts as a black body, what would be the temperature of the star, in which the rate of energy production is $Q$? (2012 Pre)
Rate of energy production $Q = P = \sigma A T^4 = \sigma (4\pi R^2)T^4$. Therefore, $T = (\frac{Q}{4\pi R^2\sigma})^{1/4}$.
Question 153:
moderateIf 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, $E' \propto (2T)^4 = 16 T^4 = 16E$.
Question 154:
moderateThe 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 $I = \frac{P}{4\pi R^2} = \frac{\sigma (4\pi r^2) T^4}{4\pi R^2} = \frac{\sigma r^2 T^4}{R^2}$.
Question 155:
A black body at $227^{\circ}\text{C}$ radiates heat at the rate of $7 \text{ cal}/\text{cm}^2 \text{ s}$. At a temperature of $727^{\circ}\text{C}$, the rate of heat radiated in the same units will be (2009)
$E \propto T^4 \Rightarrow \frac{E_2}{E_1} = (\frac{T_2}{T_1})^4 = (\frac{727+273}{227+273})^4 = (\frac{1000}{500})^4 = 16$. So, $E_2 = 16 \times 7 = 112 \text{ cal}/\text{cm}^2 \text{ s}$.
Question 156:
Assuming the sun to have a spherical outer surface of radius $r$, radiating like a black body at temperature $t^{\circ}\text{C}$, the power received by a unit surface, (normal to the incident rays) at a distance $R$ from the center of the sun is where $\sigma$ is the Stefan’s constant: (2007)
Power received by a unit surface is Intensity $I = \frac{\sigma (4\pi r^2) T^4}{4\pi R^2} = \frac{\sigma r^2 (t+273)^4}{R^2}$.
Question 157:
moderateTwo 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.
Question 158:
moderateIn 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.
Question 159:
moderateA 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 $208.7 \text{ J}$.
Question 160:
moderate11. 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.