Suppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 x 10⁵ Pa (a gauge pressure of just under 90.0 lb/in²) at a temperature of 18.0°C. What is the pressure after its temperature has risen to 35.0°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 P₀=7.00 x 10⁵ Pa, the initial temperature T₀ = 18.0°C, and the final temperature T_f = 35.0°C. We must find the final pressure P_f. How can we use the equation PV = 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: P₀V₀ = NkT₀ and P_fV_f = NkT_f. If we divide P_fV_f by P₀V₀ we can come up with an equation that allows us to solve for P_f.

Since the volume is constant, V_f and V₀ are the same and they cancel out. The same is true for N_f and N₀, and k, which is a constant. Therefore,

We can then rearrange this to solve for P_f:

where the temperature must be in units of kelvins, because T₀ and T_f are absolute temperatures.

Transcribed Image Text:

PfVf Po Vo NfkTf NokTo