Question

Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: \(PV = C\).

(a). Find the rate of change of volume with respect to pressure.

(b). A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain.

(c). Prove that the isothermal compressibility (see Example 5) is given by \(\beta = \frac{1}{P}\).

Short answer

(a). \(\frac{{dV}}{{dP}} = - \frac{C}{{{P^2}}}\)

(b). The volume is decreasing more rapidly at the beginning of the 10 minutes.

(c). Proved: \(\beta = \frac{1}{P}\)

Step-by-step solution

Boyle’s Law

The Boyle’s Lawstates that when the physical parameter such as temperature is kept constant, then thevolumeof the gas has inverseproportionality relation with the pressure exerted by it.

Evaluating the given function using differentiation:

(a)

Thegiven function is:

\(\begin{array}{r}PV = C\\ \Rightarrow V = \frac{C}{P}\end{array}\)

Now, differentiating with respect to P:

\(\begin{aligned}\frac{{dV}}{{dP}} &= \frac{d}{{dP}}\left( {\frac{C}{P}} \right)\\ &= - \frac{C}{{{P^2}}}\end{aligned}\)

Hence, this is the required answer.

Examining the obtained relation:

(b)

When the gas is being compressed steadily, then the pressure increases gradually.

Thus, the relation obtained in part (a) shows that \(\frac{{dV}}{{dP}}\) will decrease in this case.

Hence, the volume is decreasing more rapidly at the beginning of the 10 minutes.

Relation for Isothermal Compressibility: 

(c)

Now we know that:

\(\beta = - \frac{1}{V}\frac{{dV}}{{dP}}\)

Solving using part (a), we get:

\(\begin{aligned}\beta &= - \frac{1}{V}\frac{{dV}}{{dP}}\\ &= - \frac{1}{V}\left( { - \frac{C}{{{P^2}}}} \right)\\ &= \frac{C}{{P\left( {PV} \right)}}\\ &= \frac{C}{{CP}}\\ &= \frac{1}{P}\end{aligned}\)

Hence proved