​​The volume, V, of a gas at a constant temperature is inversely proportional to the
pressure, P. When the pressure is 100 N m–2, the volume is 8 litres.
(a) What happens to the volume if the pressure is doubled?
(b) What happens to the volume if the pressure in increased by a factor of 4?
(c) Find the relationship between volume and pressure.
(d) Find the pressure that gives a volume of 10 litres.

Dear Student,

Please find below the solution to the asked query:

Given : The volume, V, of a gas at a constant temperature is inversely proportional to the pressure, P.

So,

$$\begin{gathered} \mathrm{V} \propto \frac{1}{\mathrm{P}} \\ \mathbf{\Rightarrow }\mathbf{V}\mathbf{=}\frac{\mathbf{T}}{\mathbf{P}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{1}\mathbf{\hspace{0.17em}}\mathbf{)} \end{gathered}$$
As here , we have constant temperature = T

When the pressure is 100 N m², the volume is 8 litres. Substitute that value in equation 1 we get :

$$\begin{gathered} 8= \frac{\mathrm{T}}{100} \\ \mathbf{\Rightarrow }\mathbf{T}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{800}\mathbf{ }\mathbf{ }\mathbf{unit} \end{gathered}$$

a ) If the pressure is doubled :

$$\begin{gathered} {\mathrm{V}}_1 = \frac{\mathrm{T}}{2 \mathrm{P}} \\ \Rightarrow {\mathrm{V}}_1 = \frac{1}{2}\times \frac{\mathrm{T}}{\mathrm{P}} \\ \Rightarrow {\mathrm{V}}_1 = \frac{\mathrm{V}}{2} , \mathrm{From} \mathrm{equation} 1 \end{gathered}$$

If P is doubled V became half of it's initial value .                                      ( Ans )

b ) If the pressure in increased by a factor of 4 ?

$$\begin{gathered} {\mathrm{V}}_1 = \frac{\mathrm{T}}{4x \mathrm{P}} \\ \Rightarrow {\mathrm{V}}_1 = \frac{1}{4x}\times \frac{\mathrm{T}}{\mathrm{P}} \\ \Rightarrow {\mathrm{V}}_1 = \frac{\mathrm{V}}{4 x} , \mathrm{From} \mathrm{equation} 1 \end{gathered}$$

If the pressure in increased by a factor of 4 , Then V is decrease by a factor of 4 .

c ) We substitute value of T in equation 1 and get :

$\mathrm{V} = \frac{800}{\mathrm{P}}$                                                         ( Ans )


d ) Volume of 10 litres. substitute that in equation 1 and value of T = 800 ( As we calculated )  and get :

$$\begin{gathered} 10 = \frac{800}{\mathrm{P}} \\ \Rightarrow \mathrm{P} = \frac{800}{10} \\ \mathbf{\Rightarrow }\mathbf{P}\mathbf{ }\mathbf{=}\mathbf{80}\mathbf{ }\mathbf{N}\mathbf{ }{\mathbf{m}}^{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$


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