Question

Q) An automobile tire has a volume of 0.0150 m³ on a cold day when the temperature of the air in the tire is 5.0°C and atmospheric pressure is 1.02 atm. Under these conditions the gauge pressure is measured to be 1.70 atm (about 25 lb/in²). After the car is driven on the highway for 30 min, the temperature of the air in the tires has risen to 45.0°C and the volume has risen to 0.0159 m³. What then is the gauge pressure?

Short answer

The gauge pressure of the automobile tire is 1.915 atm.

Step-by-step solution

Step 1:

Ideal - gas equation on a molecular basis:

$\mathit{p}\mathit{V}\mathbf{=}\mathit{n}\mathit{R}\mathit{T}$

p = the pressure of the gas, V = the volume of the gas, n = the number of molecules of the gas, T = temperature of gas, R = 8.314 J /mol.

Step 2:

Let total initial pressure acting on the automobile tires be$p_1=1.7+1.2=2.72atm$Let total final pressure acting on the automobile tires be$p_2$.

According to Ideal gas equation

$$\begin{gathered} pV=nRT \\ p=\frac{nRT}{V} \\ \frac{P_2}{P_1}=\frac{{\displaystyle \frac{nR{T}_2}{V_2}}}{\frac{nR{T}_1}{V_1}}=\frac{T_2{V}_1}{T_1{V}_2} \\ {p}_2=\frac{T_2{V}_1}{T_1{V}_2}{p}_1 \\ =\frac{\left(273.15+45\right)\left(0.0150m^3\right)}{\left(273.15+5\right)\left(0.0159m^3\right)}\left(2.72atm\right) \\ =2.935atm \end{gathered}$$

Step 3:

Gauge pressure is a relative pressure measurement which measures pressure relative to atmospheric pressure and is defined as the absolute pressure minus the atmospheric pressure.

$$\begin{gathered} p_{gauge}=p_2-p_{atmosphere} \\ =2.935atm-1.02atm \\ =1.915atm \end{gathered}$$

Thus, the gauge pressure is 1.915 atm.