Question

Specific heat of a monatomic gas at constant volume is \[315\;J\;k{g^{ - 1}}{K^{ - 1}}\;\]and at a constant pressure is \[525\;J\;k{g^{ - 1}}{K^{ - 1}}\].Calculate the molar mass of the gas.

Answer 1

Hint: Heat Capacity is defined as amount of heat required to raise the temperature of a substance by \[1\] degree. Specific heat or specific heat capacity is the heat capacity of \[1\] kg of substance. Hence, in other words specific heat capacity is equal to heat capacity divided by mass of substance or heat capacity is product of specific heat capacity and mass of substance.

Formula Used: \[C_p{\text{ }} - {\text{ }}C_v{\text{ }} = {\text{ }}nR\]
Where C_p is the heat capacity at constant pressure and C_v is the heat capacity at constant volume. R is the universal gas constant and n is the number of moles.

Complete Step by Step Solution:
 Given: Specific heat of a monoatomic gas at constant pressure =\[525\;J\;k{g^{ - 1}}{K^{ - 1}}\]
Thus, heat capacity at constant pressure (Cp) =$525M$\[J\;{K^{ - 1}}\;\] where M is the molar mass of the gas
Specific heat of a monoatomic gas at constant volume =\[315\;J\;k{g^{ - 1}}{K^{ - 1}}\;\]
Thus, heat capacity at constant volume (Cv) =$315M$\[J\;{K^{ - 1}}\;\] where M is the molar mass of the gas
Using \[Cp{\text{ }} - {\text{ }}Cv{\text{ }} = {\text{ }}nR\]
Or For an ideal gas, the equation can be converted to \[Cp - Cv{\text{ }} = {\text{ }}\frac{R}{M}\]
Therefore, $525 - 315 = \frac{{8.314}}{M}$
$M = \frac{{8.314}}{{525 - 315}}$
$M = \frac{{8.314}}{{210}}$
\[M = 0.0396\] kg mol^-1
$M = 39.6$ g mol^-1
Hence, the correct answer is 39.6 g mol^-1

Note: There are various values of R (gas constant) such as \[8.314{\text{ }}Jmo{l^{ - 1}}{k^ - }^1\], \[2{\text{ }}cal{\text{ }}mo{l^{ - 1}}{k^{ - 1}}\],\[0.082{\text{ }}L.atm{\text{ }}mo{l^{ - 1}}{k^{ - 1}}\]. These values are used according to given units of other quantities. The above relations can be derived using the unitary method or by use of units also.