In ideal condition, the maximum efficiency that can be derived from a heat engine operating between $600\text{ K}$ reservoir and $200\text{ K}$ sink, is
Efficiency is given by $\eta = 1 - \frac{T_{\text{sink}}}{T_{\text{source}}}$. Substituting the given values: $\eta = 1 -\frac{200}{600} =\frac{2}{3} \approx 66.67%$.
The equation of state for \(14\text{ g}\) nitrogen gas at a pressure \(P\) and temperature \(T\), when occupying a volume \(V\) will be
The molecular mass of nitrogen gas \((\text{N}_2)\) is \(28\text{ g/mol}\). The number of moles is \(n = \frac{14}{28} = 0.5\). Thus, using \(PV = nRT\), we get \(PV = \frac{1}{2}RT\).
The unit of emissive power is
Emissive power is defined as the thermal energy emitted per unit area per unit time, so its unit is \(\text{J m}^{-2}\text{ s}^{-1}\) (or \(\text{W m}^{-2}\)).
Consider a sample of \(n\) moles of rigid diatomic gas. Match the columns and tick the correct option (symbols have their usual meanings):
| Column I | Column II |
| (A) Total translational kinetic energy | (P) $\frac{5}{2}K_B T$ |
| (B) Total rotational kinetic energy | (Q) $nRT$ |
| (C) Total kinetic energy per mole | (R) $\frac{3}{2}nRT$ |
| (D) Total kinetic energy per molecule | (S) $\frac{5}{2}RT$ |
Translational KE of \(n\) moles is \(\frac{3}{2}nRT\) (R). Rotational KE is \(nRT\) (Q). KE per mole of diatomic gas is \(\frac{5}{2}RT\) (S). KE per molecule is \(\frac{5}{2}k_B T\) (P). Thus, (A)-(R), (B)-(Q), (C)-(S), (D)-(P).
Two rods one made of material A and other made of material B of same length and same cross-sectional area are joined together. If thermal conductivity of material A is \(K_1\) while that of material B is \(K_2\) and the free end of rod made of material A is maintained at \(T_1\) while that of the rod of material B is maintained at \(T_2\), then the temperature of junction is (Where \(T_1 > T_2\))
Under steady state, the rate of heat flow is the same through both rods: \(\frac{K_1 A(T_1 - T_j)}{L} = \frac{K_2 A(T_j - T_2)}{L}\). Solving for \(T_j\) gives \(T_j = \frac{K_1 T_1 + K_2 T_2}{K_1 + K_2}\).
Consider the following statements out of which one is labelled as assertion and other as reason.
Assertion: The internal energy of an ideal monoatomic gas enclosed in a container does not change when there is no change in temperature.
Reason: Internal energy of a gaseous system is path function.
Internal energy of an ideal gas depends only on its temperature, so the Assertion is true. However, internal energy is a state function, not a path function, so the Reason is false.
A flask contains hydrogen and oxygen gas in the ratio of 3 : 1 by mass at temperature 27°C. The ratio of average translational kinetic energy per molecule of hydrogen and oxygen respectively is
The average translational kinetic energy per molecule of any gas is given by \(\frac{3}{2} k_B T\). Since both gases are at the same temperature, the ratio of their translational kinetic energies is 1 : 1.
Match the columns and tick the correct option. (Symbols have their usual meanings)
\begin{array}{|l|l|}
\hline
\textbf{Column-I} & \textbf{Column-II} \\ \hline
\text{a. } \gamma = \frac{5}{3} & \text{(i) Diatomic gas} \\ \hline
\text{b. } C_v = \frac{5}{2}R & \text{(ii) Triatomic non-linear gas} \\ \hline
\text{c. } C_v = 3R & \text{(iii) Monoatomic gas} \\ \hline
\end{array}
Monoatomic gas has \(\gamma = 5/3\) (a-iii). Diatomic gas has \(C_v = 5/2 R\) (b-i). Triatomic non-linear gas has \(C_v = 3R\) (c-ii). Thus, the correct matching is a(iii), b(i), c(ii).
Four moles of helium are mixed with two moles of oxygen. The molar specific heat capacity of the mixture at constant volume is
For helium (monoatomic), \(C_{v1} = \frac{3}{2}R\) and \(n_1 = 4\). For oxygen (diatomic), \(C_{v2} = \frac{5}{2}R\) and \(n_2 = 2\). The mixture molar specific heat is \(C_{v,\text{mix}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} = \frac{4 \times 1.5R + 2 \times 2.5R}{4 + 2} = \frac{11R}{6}\).
In winters, a metal surface feels cooler upon touching than a wooden surface because
Metal is a much better conductor of heat than wood. When touched in winters, heat is rapidly conducted away from our hand to the metal surface, making it feel colder.