Answer to Question #91958 in Molecular Physics | Thermodynamics for Raghav arora

Question

4 gm of a gas occupy 22.4 litres at into . The specific heat of gas at constant vol is 5 j/k .if speed of any quantity x of gas is at into is 952m/s. Then heat capacity at constant pressure is(R=8.3j/k/mol)

Answer 1

The volume of a mole of any ideal gas at NTP (normal temperature and pressure) is 22.4 L. Since 4.0 g of a gas occupies 22.4 liters at NTP, so the molecular mass of the gas is M=4.0 g mol^-1=4.0·10^-3 kg mol^-1.

The speed of the sound in the gas is

$$v=\sqrt{\frac{\gamma RT}{M}}$$

where $\gamma$ is the ratio of the heat capacities at constant pressure, C_p, and at constant volume, C_v,

$$\gamma =\frac{{{C}_{p}}}{{{C}_{v}}}$$

R=8.3 J K^-1 mol^-1 is the universal gas constant, T is the temperature of the gas (T=273 K at NTP). We are also given that v=952 m s^-1.

Find $\gamma$ :

$${{v}^{2}}=\frac{\gamma RT}{M}\,\,\,\Rightarrow \,\,\,\gamma =\frac{M{{v}^{2}}}{RT}$$

Substitute known values

$$\gamma =\frac{4.0\cdot {{10}^{-3}}\text{ kg mo}{{\text{l}}^{-1}}\cdot {{\left( 952\,\text{m}\,{{\text{s}}^{-1}} \right)}^{2}}}{8.3\text{ J }{{\text{K}}^{-1}}\text{mo}{{\text{l}}^{-1}}\cdot 273\,\text{K}}=1.6$$

Now find heat capacity at constant pressure

$$\gamma =\frac{{{C}_{p}}}{{{C}_{v}}}\,\,\,\Rightarrow \,\,\,{{C}_{p}}=\gamma {{C}_{v}}$$

Since $\gamma$ =1.6 and C_v=5 J K^-1mol^-1, then

$${{C}_{p}}=1.6\cdot 5\text{ J }{{\text{K}}^{-1}}\text{mo}{{\text{l}}^{-1}}=8\,\text{J }{{\text{K}}^{-1}}\text{mo}{{\text{l}}^{-1}}$$