13.3 The ideal gas law  (Page 2/11)

Calculating pressure changes due to temperature changes: tire pressure

Suppose your bicycle tire is fully inflated, with an absolute pressure of $7\text{.}\text{00}\times {\text{10}}^5\text{Pa}$ (a gauge pressure of just under $\text{90}\text{.}0{\text{lb/in}}^2$ ) at a temperature of $\text{18}\text{.}0\text{º}\text{C}$ . What is the pressure after its temperature has risen to $\text{35}\text{.}0\text{º}\text{C}$ ? Assume that there are no appreciable leaks or changes in volume.

Strategy

The pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.

We know the initial pressure $P_0=7\text{.00}\times {\text{10}}^5\text{Pa}$ , the initial temperature $T_0=\text{18}\text{.}\mathrm{0ºC}$ , and the final temperature $T_{\text{f}}=35\text{.}\mathrm{0ºC}$ . We must find the final pressure $P_{\text{f}}$ . How can we use the equation $\text{PV}=\text{NkT}$ ? At first, it may seem that not enough information is given, because the volume $V$ and number of atoms $N$ are not specified. What we can do is use the equation twice: $P_0{V}_0={\text{NkT}}_0$ and $P_{\text{f}}{V}_{\text{f}}={\text{NkT}}_{\text{f}}$ . If we divide $P_{\text{f}}{V}_{\text{f}}$ by $P_0{V}_0$ we can come up with an equation that allows us to solve for $P_{\text{f}}$ .

Since the volume is constant, $V_{\text{f}}$ and $V_0$ are the same and they cancel out. The same is true for $N_{\text{f}}$ and $N_0$ , and $k$ , which is a constant. Therefore,

We can then rearrange this to solve for $P_{\text{f}}$ :

where the temperature must be in units of kelvins, because $T_0$ and $T_{\text{f}}$ are absolute temperatures.

Solution

1. Convert temperatures from Celsius to Kelvin.

2. Substitute the known values into the equation.

Discussion

The final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that absolute pressure and absolute temperature must be used in the ideal gas law.