Question

A gas is in a sealed container. By what factor does the gas pressure change if:

a. The volume is doubled and the temperature is tripled?

b. The volume is halved and the temperature is tripled?

Short answer

a. The final pressure enhance 1.5 times of initial pressure.

b. The final pressure enhance will 6 times.

Step-by-step solution

 Formulation of ideal gas

As per, the formula of ideal gas is

$\begin{array}{rcl}{P}^{\text{'}}{V}^{\text{'}}& =& nR{T}^{\text{'}}\\ P^{\text{'}}& =& \frac{nR{T}^{\text{'}}}{V^{\text{'}}}\\ & & \end{array}$

$\begin{array}{rcl}& & P^{\text{'}},V^{\text{'}},T^{\text{'}}\end{array}$are initial pressure, volume, and temperature.

Let $\begin{array}{rcl}& & P^{\text{'}\text{'}},V^{\text{'}\text{'}},T^{\text{'}\text{'}}\end{array}$are final pressure, volume, and temperature.

Applying the

role="math" localid="1648462022127" $$\begin{gathered} V^{\text{'}\text{'}}=2V^{\text{'}} \\ {T}^{\text{'}\text{'}}=3T^{\text{'}} \end{gathered}$$

Then put the value of above equation

role="math" localid="1648462325734" $\begin{array}{rcl}& & \\ P^{\text{'}\text{'}}& =& \frac{nR{T}^{\text{'}\text{'}}}{V^{\text{'}\text{'}}}\\ & =& \frac{nR3T^{\text{'}}}{2V^{\text{'}}}\\ P^{\text{'}\text{'}}& =& \frac{3}{2}{P}^{\text{'}}\\ P^{\text{'}\text{'}}& =& 1.5P^{\text{'}}\end{array}$

So, the final pressure enhance 1.5 times of initial pressure.

Equation of ideal gas

Assume

$$\begin{gathered} V^{\text{'}\text{'}}=\frac{1}{2}{V}^{\text{'}} \\ {T}^{\text{'}\text{'}}=3T^{\text{'}} \end{gathered}$$

Applying all data

$\begin{array}{rcl}& & \\ P^{\text{'}\text{'}}& =& \frac{nR{T}^{\text{'}\text{'}}}{V^{\text{'}\text{'}}}\\ & =& \frac{nR3T^{\text{'}}}{{\displaystyle \frac{1}{2}}{V}^{\text{'}}}\\ P^{\text{'}\text{'}}& =& 6P^{\text{'}}\\ & & \end{array}$

So, the final pressure enhance will 6 times.