Numerical 1

Air at 273 K and 1.01 [latex]\times[/latex] 10⁵ Nm^-2 pressure contains 2.70 [latex]\times [/latex] 10²⁵ molecules per cubic meter. How many molecules per cubic meter will there be at a place, where the temperature is 223K and the pressure is 1.33 [latex]\times[/latex] 10⁴ Nm^-2?

Solution

Given,

Temperature (T₁) = 273 K

Pressure (P₁) = 1.01 [latex]\times[/latex] 10⁵ Nm^-2

No. of molecules per unit volume (n₁) = 2.70 [latex]\times[/latex] 10²⁵ molecules/m³

No. of molecules per unit volume at T₂ (n₂) =?

Temperature (T₂) = 223 K

Pressure (P₂) = 1.33 [latex]\times[/latex] 10⁴ Nm^-2

We know that,

P = [latex]\frac{1}{3}\rho c^2[/latex]

Or, P = [latex]\frac{1}{3}\times \frac{M}{V}\times c^2[/latex]

Or, P = [latex]\frac{1}{3}\times \frac{mN}{V}\times c^2[/latex]