Question

0.0500 mol of gas occupies a cylinder which is sealed on top by a moveable piston. The piston is circular, with a mass of 30.0 kg and diameter of 5.00 cm. It is supported only by the pressure of the gas in the cylinder. Outside of the cylinder, there is air at 1.00 atm. Initially, the piston is 30.0 cm above the bottom of the cylinder. Heat is then added, causing the gas to expand until the piston is 40.0 cm above the bottom of the cylinder. For this process, find the change in internal energy, the work done by the gas, and the heat which flows into the gas.

a. assuming that the gas is N2.
b. assuming that the gas is neon

Answer 1

The workdone by both N₂ and neon gas is 49.3 J

The change in internal energy of N₂ and neon gas is 125.6 J and 73.54 J respectively

The heat for N₂ and neon gas is 171.9 J and 122.84 J respectively.

Step-by-step explanation:

Given that:

number of moles = 0.05 mole

mass of the piston = 30 kg

diameter = 5.00 cm = 0.05 m

Area (A) = πr²

`Area (A) = \pi*((0.05)/(2))^2 \\ \\ Area (A) = 0.0019635 \\ \\ Area (A) = 19.635*10^(-4) \ m^2`

The piston is said to move from 30 cm - 40 cm

So, the change in volume ΔV is calculated as:

`=(40-30)*10^(-2) *19.635*10^(-4)`

`= 1.9635*10^(-4) \ m^3`

Outside the cylinder; the pressure
`P_(air)= 1 \ atm` = 101325 Pa

Thus, workdone
`w_1` = PΔV

=
`101325*1.9635*10^(-4)`

= 19.90 J

The gravitational work
`w_2 = mgh`

Given that the height (h) = 10 cm = 0.1 m

Then;
`w_2 = 30*9.8*0.1`

`w_2 = 29.4 \ J`

The total workdone
`w_(total)` for both cases is:

`w_(total ) =w_1 + w_2`

`w = (19.90 + 29.4) \ J`

`w =49.3 \ J`

The pressure of gas inside the cylinder is determined as:

`P_(in).A = P_(out).A +mg`

`(P_(in)-P_(out)) = (mg)/(A) \\ \\ P_(in) -10^5 = (30*9.8)/(19.635*10^(-4)) \\ \\ P_(in) = 149732.6203+10^5 \\ \\ P_(in) = 2.497*10^5 \ Pa`

a). assuming that the gas is N₂.

`C_v =(5)/(2)R`

Thus, the change in internal energy ΔU is given as:

`\delta U = nC_v \delta T \\ \\ \delta U = n* (5)/(2)R \delta T \\ \\ \delta U = (5)/(2)nR \delta T`

Since
`P_(in) \delta V = nR \delta T ; \ Then`;

`\delta \ U = (5)/(2) P_(in) \delta V \\ \\ \delta \ U = (5)/(2)*2.497*10^5 *1.9635*10^(-4) \\ \\ \delta \ U = 122.57 \ J`

ΔU ≅ 125.6 J

The heat Q = ΔU + W

Q = (122.6 + 49.3) J

Q = 171.9 J

b) In Neon gas:

`C_v = (3)/(2)R`

∴

change in internal energy is;

`\delta U = nC_v \delta T \\ \\ \delta U = n* (3)/(2)R \delta T \\ \\ \delta U = (3)/(2)P_(in).V`

`\delta U = (3)/(2)*2.497*10^5*1.9635*10^(-4)`

ΔU = 73.54 J

The heat Q = ΔU + W

Q = (73.54 + 49.3) J

Q = 122.84 J