Question 181:
moderateA Carnot engine whose sink is at $300\text{ K}$ has an efficiency of $40\%$. By how much should the temperature of source be increased so as to increase its efficiency by $50\%$ of original efficiency? (2006)
Initial source temperature $T_1 = \frac{300}{1-0.4} = 500\text{ K}$. New efficiency $\eta' = 1.5 \times 0.4 = 0.6$. New source temperature $T_1' = \frac{300}{1-0.6} = 750\text{ K}$. Increase $\Delta T = 750 - 500 = 250\text{ K}$.
Question 182:
moderateAn ideal gas heat engine operates in Carnot cycle between $227\text{ }^\circ\text{C}$ and $127\text{ }^\circ\text{C}$. It absorbs $6 \times 10^4\text{ cal}$ of heat at higher temperature. Amount of heat converted to work is: (2005)
Temperatures are $T_1 = 500\text{ K}$ and $T_2 = 400\text{ K}$. Efficiency $\eta = 1 - \frac{400}{500} = 0.2$. Work $W = \eta Q_1 = 0.2 \times 6 \times 10^4 = 1.2 \times 10^4\text{ cal}$.
Question 183:
moderateAn ideal gas heat engine operates in a Carnot cycle. Between $227\text{ }^\circ\text{C}$ and $127\text{ }^\circ\text{C}$. It absorbs $6\text{ kcal}$ at the higher temperature. The amount of heat (in kcal) converted into work is equal to: (2003)
Efficiency $\eta = 1 - \frac{400}{500} = 0.2$. Work done $W = \eta Q_1 = 0.2 \times 6\text{ kcal} = 1.2\text{ kcal}$.
Question 184:
moderateThe efficiency of carnot engine is $50\%$ and temperature of sink is $500\text{ K}$. If temperature of source is kept constant and its efficiency raised to $60\%$, then the required temperature of the sink will be: (2002)
Source temperature $T_1 = \frac{500}{1-0.5} = 1000\text{ K}$. New sink temperature $T_2' = T_1(1-\eta_2) = 1000(1-0.6) = 400\text{ K}$.
Question 185:
moderateA scientist says that the efficiency of his heat engine which work at source temperature $127\text{ }^\circ\text{C}$ and sink temperature $27\text{ }^\circ\text{C}$ is $26\%$, then: (2001)
Maximum Carnot efficiency $\eta_{\text{max}} = 1 - \frac{300}{400} = 25\%$. Since the claimed efficiency ($26\%$) exceeds the Carnot limit, it is impossible.
Question 186:
moderateThe ratio ($W/Q$) for a carnot-engine is $1/6$. Now the temperature of sink is reduced by $62\text{ }^\circ\text{C}$, this ratio becomes twice, therefore the initial temp. of the sink and source are respectively: (2000)
Ratio $W/Q_1$ is the efficiency $\eta_1 = 1/6$. When sink temperature decreases, efficiency becomes $2/6 = 1/3$. Solving yields source $T_1 = 372\text{ K} = 99\text{ }^\circ\text{C}$ and sink $T_2 = 310\text{ K} = 37\text{ }^\circ\text{C}$.
Question 187:
moderateThe efficiency of a Carnot engine operating with reservoir temperature of $100\text{ }^\circ\text{C}$ and $-23\text{ }^\circ\text{C}$ will be: (1997)
Efficiency $\eta = 1 - \frac{T_2}{T_1} = 1 - \frac{250\text{ K}}{373\text{ K}} = \frac{373-250}{373}$.
Question 188:
moderateAn ideal Carnot engine, whose efficiency is $40\%$, receives heat at $500\text{ K}$. If its efficiency is $50\%$, the intake temperature for the same exhaust temperature is: (1995)
Exhaust temperature $T_2 = 500(1-0.4) = 300\text{ K}$. New intake temperature $T_1' = \frac{300}{1-0.5} = 600\text{ K}$.
Question 189:
The molar specific heats of an ideal gas at constant pressure and volume are denoted by $C_p$ and $C_v$ respectively. If $\gamma = \frac{C_p}{C_v}$ and R is the universal gas constant, then $C_v$ is equal to: (2013)
From Mayer's relation, $C_p - C_v = R$. Given $\gamma = \frac{C_p}{C_v} \implies C_p = \gamma C_v$. Substituting, $\gamma C_v - C_v = R \implies C_v(\gamma - 1) = R \implies C_v = \frac{R}{\gamma - 1}$.
Question 190:
To find out degree of freedom, the correct expression is: (2000)
The relation between the ratio of specific heats $\gamma$ and degrees of freedom $f$ is given by $\gamma = 1 + \frac{2}{f}$. Rearranging this, we get $\frac{2}{f} = \gamma - 1 \implies f = \frac{2}{\gamma - 1}$.