GSO002 Problem 6

Problem Statement

A cyclist uses a cylinder-type pump to inflate a bicycle tire. Consider the compression process while the inside of the cylinder is sealed (closed system) — from when the piston begins to be pushed in until just before the tire valve opens and air begins to enter the tire.

Treat the air inside the cylinder as a single thermodynamic system, modeled as an ideal gas with constant-volume molar heat capacity $C_V = \frac{5}{2}R$CV​=25​R ($R$R is the gas constant).

In the initial state, the air inside the cylinder has pressure $P_1$P1​ and volume $V_1$V1​. After the cyclist pushes the piston in, the air is compressed, and just before the valve opens, the pressure is $P_2$P2​ and the volume is $V_2$V2​.

Let $W_{in}$Win​ be the work done on the gas by the cyclist (external agent) during this compression. Due to the temperature rise caused by compression, the air releases a heat $Q_{out}$Qout​ to the outside through the cylinder wall.

Find the heat $Q_{out} [\mathrm{J}]$Qout​[J] released by the air to the outside during this process.

Sign conventions:

• $P_1, V_1, P_2, V_2, W_{in}, Q_{out}$P1​,V1​,P2​,V2​,Win​,Qout​ are all positive.
• With $Q$Q = heat absorbed by gas from outside, $W$W = work done by gas on outside, $\Delta U$ΔU = change in internal energy: the first law is $Q = \Delta U + W$Q=ΔU+W.

Constraints

• Initial pressure: $P_1 = 1.0 \times 10^5\ \mathrm{Pa}$P1​=1.0×105 Pa
• Initial volume: $V_1 = 2.5 \times 10^{-3}\ \mathrm{m}^3$V1​=2.5×10−3 m3
• Final pressure (just before valve opens): $P_2 = 3.0 \times 10^5\ \mathrm{Pa}$P2​=3.0×105 Pa
• Final volume (just before valve opens): $V_2 = 1.48 \times 10^{-3}\ \mathrm{m}^3$V2​=1.48×10−3 m3
• Work done on gas by external agent: $W_{in} = 562\ \mathrm{J}$Win​=562 J

Input Format

Give the value of $Q_{out} [\mathrm{J}]$Qout​[J] as a positive integer.