A thermally insulated container is divided into two parts by a thermally insulat

Question

A thermally insulated container is divided into two parts by a thermally insulated partition. Both parts contain ideal gases which has equal constant heat capacities C_V. One of these parts contains v_1 moles of gas at a temperature T_1 and pressure p_1; the other contains v_2 moles of gas at a temperature T_2 and pressure p_2. The partition is now removed and the system is allowed to come to equilibruim.

(a) Find the final pressure?

(b) Find the change (delta)S of total entropy if the gases are different

(c) Find (delta)S if the gases are identical

Explanation / Answer

a) We use the law of Dalton to know the final pressure, and it is:

P_m=P_1+P_2, P is the final pressure for both gases when they combine

b) The change of entropy when they are different:

first we need to know all the properties of the gases when they combine, so:

v_m=v_1+v_2 is the number of moles of the 2 gases when they combine

the volume of the two divided gases are:

V_1=(T_1 x R x v_1) / P_1 the volume of the gas 1

V_2=(T_2 x R x v_2)/ P_2 the volum of the gas 2, where R is the constant of gases

so the volume of the gas combine is: V_1+V_2= V_m, and the temperature is:

T_m=(P_m x V_m) / ( R x v_m) and whe have all this information

so the change of each entropy is: S_1= C_v x ln(T_m/T_1) + R x ln (V_m/V_1)

and S_2= C_v x ln(T_m / T_2) + R x ln(V_m/V_2)

and the change is: S_m=S_1+S_2= C_v x ln(T_m / T_2) + R x ln(V_m/V_2) + C_v x ln(T_m/T_1) + R x ln (V_m/V_1)

c) If they are the same is:

S_m=2 x S_1 =2x [C_v x ln(T_m/T_1) + R x ln (V_m/V_1) ]

Good luck!

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