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
AIPMT 2013,AIEEE 2003, Medium
Download our app for free and get started
$P \propto {T^3};\,\,\,\,\,\,P{T^{ - 3}}=constant$ $...(i)$
For an adiabatic process; $P{T^{\frac{\gamma }{{1 - \gamma }}}}=constant$ $...(ii)$
Comparing $(i)$ and $(ii)$, we get
$\frac{\gamma }{{1 - \gamma }} = - 3\,\,;\,\,\gamma = - 3 + 3\gamma $
$ - 2\gamma = - 3\,\,or\,\,\gamma = \frac{3}{2}$
$As\,\,\,\gamma = \frac{{{C_p}}}{{{C_v}}}\,\,\,\therefore \,\,\,\frac{{{C_p}}}{{{C_v}}} = \frac{3}{2}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A hypothetical gas expands adiabatically such that its volume changes from $8$ litres to $27$ litres. If the ratio of final pressure of the gas to initial pressure of the gas is $\frac{16}{81}$. Then the ratio of $\frac{C_P}{C_V}$ will beView Solution
- 2An engine operates by taking $n\,moles$ of an ideal gas through the cycle $ABCDA$ shown in figure. The thermal efficiency of the engine is : (Take $C_v =1 .5\, R$, where $R$ is gas constant)View Solution
- 3Two carnot engines $A$ and $B$ operate in series such that engine $A$ absorbs heat at $T_{1}$ and rejects heat to a sink at temperature $T$. Engine $B$ absorbs half of the heat rejected by engine $A$ and rejects heat to the sink at ${T}_{3}$. When workdone in both the cases is equal, the value of ${T}$ isView Solution
- 4A Carnot engine takes $6000 \,cal$ of heat from a reservoir at $627^{\circ} C$ and gives it to a sink at $27^{\circ} C$. The work done by the engine is ......... $kcal$View Solution
- 5A mixture of gases at $STP$ for which $\gamma=1.5$ is suddenly compressed to $\frac{1}{9}$ th of its original volume. The final temperature of mixture is .......... $^{\circ} C$View Solution
- 6The pressure and volume of a gas are changed as shown in the $P-V$ diagram in this figure. The temperature of the gas will ........View Solution
- 7The amount of heat needed to raise the temperature of $4\, moles$ of a rigid diatomic gas from $0^{\circ} {C}$ to $50^{\circ} {C}$ when no work is done is ......${R}$ ($R$ is the universal gas constant)View Solution
- 8A thermodynamic cycle takes in heat energy at a high temperature and rejects energy at a lower temperature. If the amount of energy rejected at the low temperature is $3$ times the amount of work done by the cycle, the efficiency of the cycle isView Solution
- 9During which of the following thermodynamic process represented by $P V$ diagram the heat energy absorbed by system may be equal to area under $P V$ graph?View Solution
- 10A van der Waal's gas obeys the equation of state $\left(p+\frac{n^2 a}{V^2}\right)(V-n b)=n R T$. Its internal energy is given by $U=C T-\frac{n^2 a}{V}$. The equation of a quasistatic adiabat for this gas is given byView Solution