A bicycle pump takes a volume ΔV of air at atmospheric pressure p0 and constant temperature T0 and compresses it so that it enters a tire that has a volume V0. The air inside the tire is initially at atmospheric pressure p0 and can be considered as an ideal gas.Determine the number of times n the

Question

A bicycle pump takes a volume ΔV of air at atmospheric pressure $p_0$ and constant temperature $T_0$ and compresses it so that it enters a tire that has a volume $V_0$ . The air inside the tire is initially at atmospheric pressure $p_0$ and can be considered as an ideal gas. Determine the number of times n the user has to pump air into the tire to reach a pressure $p_f$ . Assume that the pump is designed so that the air in the tire is always at temperature $T_0$ .

Numerical Application:

$V_0$ = 50 l, ΔV = 1.2 l and $p_f = 2.5 p_0$ .

Numerical Application:

[latex]V_0[/latex] = 50 l, ΔV = 1.2 l and [latex]p_f = 2.5 p_0[/latex].

Accepted Answer

The initial and final number of moles of air inside the tire of volume [latex]V_0[/latex] at temperature [latex]T_0[/latex] are given by,

[latex] N_0 = \frac{p_0 V_0}{R T_0} [/latex].

and
[latex] N_f = \frac{p_f V_0}{R T_0} [/latex] thus [latex]\frac{N_f}{N_0} = \frac{p_f}{p_0}[/latex].

The additional number of moles of air pumped into the tire each time are,
[latex] ΔN = \frac{p_0 ΔV}{R T_0} [/latex] and [latex] N_f = N_0 + n ΔN [/latex].

Thus,

[latex]\frac{N_f}{N_0} = 1+ n \frac{ΔN}{N_0} = 1+ n \frac{p_0 ΔV}{N_0 R T_0} = 1+ n \frac{ΔV}{V_0} = \frac{p_f}{p_0} [/latex].

which implies that,

[latex] n = \biggl(\frac{p_f}{p_0}-1\biggr) \frac{V_0}{\Delta V} = 62.5 [/latex].

This means that the air has to be pumped 63 times in order to reach a final pressure [latex]p_f[/latex] that is at least 2.5 [latex]p_0[/latex].

Question 5.2: A bicycle pump takes a volume ΔV of air at atmospheric press...

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