NEET-XII-Physics
24: Kinetic Theory of Gases
- #10The temperature and pressure at Simla are 15.0°C and 72.0 cm of mercury and at Kalka these are 35.0°C and 76.0 cm of mercury. Find the ratio of air density at Kalka to the air density at Simla.Ans : Here,
Temperature in Simla, T1= 15 + 273 = 288 K
Pressure in Simla, P1 = 0.72 m of Hg
Temperature in Kalka, T2= 35+273 = 308 K
Pressure in Kalka, P2 = 0.76 m of Hg
Let density of air at Simla and Kalka be `` \rho ``1 and `` \rho ``2, respectively. Then,
`` PV=\frac{\mathit{m}}{\mathit{M}}RT``
`` \Rightarrow \frac{\mathit{m}}{\mathit{V}}\mathit{=}\frac{\mathit{P}\mathit{M}}{\mathit{R}\mathit{T}}``
`` \Rightarrow \rho \mathit{=}\frac{\mathit{P}\mathit{M}}{\mathit{R}\mathit{T}}``
Thus,
`` {\rho }_{1}=\frac{{P}_{1}M}{\,\mathrm{\,R\,}{T}_{1}}``
`` {\rho }_{2}=\frac{{P}_{2}M}{\,\mathrm{\,R\,}{T}_{2}}``
Taking ratios, we get
`` \frac{{\rho }_{\mathit{1}}}{{\rho }_{2}}=\frac{{P}_{1}}{{T}_{1}}\times \frac{{T}_{2}}{{P}_{2}}``
`` \Rightarrow \frac{{\rho }_{1}}{{\rho }_{2}}=\frac{0.72}{288}\times \frac{308}{0.76}``
`` \Rightarrow \frac{{\rho }_{2}}{{\rho }_{\mathit{1}}}=0.987``