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]

Or, P = [latex]\frac{1}{3}\times m \times (\frac{N}{V})\times c^2[/latex]

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

So, 1^st case:

P₁ = [latex]\frac{1}{3}mn_1 \times c^2[/latex] ……………. (i)

For 2^nd case:

P₂ = [latex]\frac{1}{3}mn_2 [/latex] ……………… (ii)

Dividing eq^n. (i) by eq^n. (ii), we get,

[latex]\frac{P_1}{P_2} = \frac{\frac{1}{3}mn_1c_1^2}{\frac{1}{3}mn_2c_2^2}[/latex]

Or, [latex]\frac{P_1}{P_2} = \frac{n_1\times T_1}{n_2\times T_2}[/latex]      [latex][\because c \propto \sqrt{T}][/latex]       

Or, n₂ = [latex]\frac{n_1\times T_1}{P_1\times T_2}\times P_2[/latex]

= [latex]\frac{(2.70\times 10^{25})\times 273}{(1.01\times 10^5)\times 223}\times (1.33\times 10^4)[/latex]

= 4.353 [latex]\times[/latex] 10²⁴ molecules/m³.

Numerical 2

The correct inflation of a tyre at 20^oC is 2 kg/cm². After driving several hours, the driver checks the tyres. If the tyre’s temperature is 50^oC, what should be the pressure reading?

Solution

Given,

Temperature (T₁) = 20 + 273 = 293 K

Pressure at T₁ (P₁) = 2kg/cm²

= [latex]2\times 10 \times 10^4[/latex] N/m²

= 2 x 10⁵ N/m²

Now, Pressure at T₂ (P₂) =?

Temperature (T₂) = 50 + 273 = 323 K

We know, at constant volume,

[latex]\frac{P_1}{T_1} = \frac{P_2}{T_2}[/latex]

Or, P₂ = [latex]\frac{P_1}{T_1}\times T_2[/latex]

= [latex]\frac{2\times 10^5}{293}\times 323[/latex]

= 2.205 x 10⁵ Nm^-2.