Question

If the cylindrical pistons are 25.000 cm in diameter at 20.0 ∘c, what should be the minimum diameter of the cylinders at that temperature so the pistons will operate at 150.0 ∘c

Answer 1

Refer to the diagram shown below.

The piston supports the same load W at both temperatures.
The ideal gas law is

`pV=nRT`
where
p = pressure
V = volume
n = moles
T = temperature
R = gas constant

State 1:
T₁ = 20 C = 20+273 = 293 K
d₁ = 25 cm piston diameter

State 2:
T₂ = 150 C = 423 K
d₂ = piston diameter

Because V, n, and R remain the same between the two temperatures, therefore

`(p_(1))/(T_(1)) = (p_(2))/(T_(2))`

If the supported load is W kg, then

`p_(1) = (W \, N)/( (\pi)/(4) d_(1)^(2)) = (4W \, N)/(\pi (0.25 \, m)^(2)) = 20.3718W \, Pa`
Similarly,

`p_(2) = (4W)/(\pi d_(2)^(2)) \, Pa`


`(p_(1))/(p_(2)) = (20.3718 \pi d_(2)^(2))/(4) = 16 d_(2)^(2)`

Because p₁/p₂ = T₁/T₂, therefore

`16d_(2)^(2) = (293)/(423) \\\\ d_(2)^(2) = (0.6927)/(16) \\\\ d_(2) = 0.2081 \, m`

The minimum piston diameter at 150 C is 20.8 cm.

Answer: 20.8 cm diameter