Question

Question #0a30f

Answer 1

(a) The pressure rise in the tyre is 149 kPa. (b) The mass of air that must be bled off is 17 g.

Explanation:

(a) Calculate the pressure rise.

Since the volume is constant but the pressure and temperature are changing, this is an example of Gay-Lussac's Law:

`color(blue)(bar(ul(|color(white)(a/a) P_1/T_1=P_2/T_2color(white)(a/a)|)))" "`

We can rearrange the above formula to get

`P_2=P_1 × T_2/T_1`

A gauge pressure (`P_"g"`) is the pressure difference between a system `P_"s"` and the surrounding atmosphere `P_"atm"`.

`P_"g" = P_"s" - P_"atm"`

or

`P_"s" = P_"g" + P_"atm"`

Your data are:

`P_1 = "(210 + 101.3) kPa = 311.3 kPa"`
`T_1 = "(250 + 273.15) K" = "523.15 K"`
`P_2 = "?"`
`T_2 = "(500 + 273.15) K" = "773.15 K"`

`P_2 = "311.3 kPa" × (773.15 color(red)(cancel(color(black)("K"))))/(523.15 color(red)(cancel(color(black)("K")))) = "460.1 kPa"`

The new pressure is 460.1 kPa.

The pressure rise is

`ΔP = P_2 - P_1 = "(460.1 - 311.3) kPa = 149 kPa"`

(b) Calculate the mass of air to be bled off.

For this calculation, we can use the Ideal Gas Law:

`color(blue)(bar(ul(|color(white)(a/a)PV = nRTcolor(white)(a/a)|)))" "`

Since

`"moles" = "mass"/"molar mass"` or `n = m/M`,

We can re-write the Ideal Gas Law as

`PV = m/MRT`

or

`m = (PVM)/(RT)`

Since `V, M, R"` and `T` are constant, we can write

`Δm = ΔP × (VM)/(RT)`

`Δm = 149 × 10^3 color(red)(cancel(color(black)("Pa"))) × (0.025 color(red)(cancel(color(black)("m"^3))) × "29 g"·color(red)(cancel(color(black)("mol"^"-1"))))/(8.314 color(red)(cancel(color(black)("Pa·m"^3"K"^"-1""mol"^"-1"))) × 773.15 color(red)(cancel(color(black)("K")))) = "17 g"`

You would have to bleed off 17 g of air.