Question #0a30f

1 Answer
Mar 1, 2017

Answer:

(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.