Question

A piston-cylinder device contains an ideal gas mixture of 3 kmol of He gas and 7 kmol of Ar gas (both gases are monatomic) at 27°C and 200 kPa. Now the gas expands at constant pressure until its volume doubles. The amount of heat transfer to the gas mixture is, in MJ (round to nearest integer; for example if the answer is 53.7MJ, write 54; if the answer is 52.1MJ write 52; do not include the units in your answer)

Answer 1

Q = 62

Step-by-step explanation:

The solution is obtained by manipulating the energy balance of the system:

Q = W + ΔU

⇒ Q = P*(V₂ - V₁) + mcv*(T₂ - T₁)

⇒ Q = P*V₁ + m*(((mcv)He /m) + ((mcv)Ar /m)*((P*V₂/(mR) - T₁)

⇒ Q = mRT₁ + ((mcv)He + (mcv)Ar)*(2P*V₁/(mR) - T₁)

⇒ Q = T₁*(Nm*Ru + (mcv)He + (mcv)Ar)

⇒ Q = T₁*(Nm*Ru + (MNcv)He + (MNcv)Ar)

where

T₁ = (27 + 273) K = 300 K

Nm = (7 + 3) kmol = 10 kmol = 10⁴ mol

Ru = 8.314 J*K⁻¹*mol⁻¹

(MNcv)He = (4 g*mol⁻¹)*(3*10³ mol)*(3.1156 kJ*Kg⁻¹*K⁻¹)*(1 kg/10³ g)*(1 MJ/10³ kJ) = 0.0373872 MJ*K⁻¹

(MNcv)Ar = (40 g*mol⁻¹)*(7*10³ mol)*(0.3122 kJ*Kg⁻¹*K⁻¹)*(1 kg/10³ g)*(1 MJ/10³ kJ) = 0.087416 MJ*K⁻¹

Finally, we get

⇒ Q = 300*(10*8.314 + 4*3*3.1156 + 40*7*0.3122)*10⁻³

⇒ Q = 62

Answer 2

Q = 62 ( since we are instructed not to include the units in the answer)

Step-by-step explanation:

Given that:

`n_(HCl) = 3 \ kmol\\n_(Ar) = 7 \ k mol`

`T_1 = 27^0 \ C = ( 27+273)K = 300 K`

`P_1 = 200 \ kPa`

Q = ???

Now the gas expands at constant pressure until its volume doubles

i.e if
`V_1 = x\\V_2 = 2V_1`

Using Charles Law; since pressure is constant

`V \alpha T`

`(V_2)/(V_1) =(T_2)/(T_1)`

`(2V_1)/(V_1) =(T_2)/(300)`

`T_2 = 300*2\\T_2 = 600`

mass of He =number of moles of He × molecular weight of He

mass of He = 3 kg × 4

mass of He = 12 kg

mass of Ar =number of moles of Ar × molecular weight of Ar

mass of He = 7 kg × 40

mass of He = 280 kg

Now; the amount of Heat Q transferred =
`m_(He)Cp_(He) \delta T + m_(Ar)Cp_(Ar) \delta T`

From gas table

`Cp_(He) = 5.9 \ kJ/Kg/K\\Cp_(Ar) = 0.5203 \ kJ/Kg/K`

∴ Q =
`12*5.19*10^3(600-300)+280*0.5203*10^3(600-300)`

Q =
`62.389 *10^6`

Q = 62 MJ

Q = 62 ( since we are instructed not to include the units in the answer)