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Calculate the values of gamma = Cp//Cv f...

Calculate the values of `gamma = C_p//C_v` for a gaseous mixture consisting of `v_1 = 2.0` moles of oxygen and `v_2 = 3.0` mole of carbon dioxide. The gases are assumed to be ideal.

Text Solution

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Total no. of moles of the mixture `v = v_1 + v_2`
At a certain temperature, `U = U_1 + U_2` or `v C_V = v_1 + v_2 C_(V_2)`
Thus `C_V = (v_1 C_(v_1) + v_2 C_(v_2))/(v) = ((v_1 (R)/(gamma_1 - 1)+v_2(R)/(gamma_2 - 1)))/(v)`
Similarly `C_p = (v_1 C_(p_1) + v_2 C_(p_2))/(v)`
`(v_1 gamma_2 C_(v_1) gamma_2 C_(v_2))/(v) = ((v_1(gamma_1 R)/(gamma_1 - 1) + v_2 (gamma_2 R)/(gamma_2 - 1)))/(v)`
Thus `gamma = (C_p)/(C_V) = (v_1(gamma_1)/(gamma_1 - 1)R + v_2(gamma_2)/(gamma_2 - 1)R)/(v_1(R)/(gamma_1 - 1) + v_2 (R)/(gamma_2 - 1))`
=`(v_1 gamma_1(gamma_2 - 1)+ v_2 gamma_2(gamma_1 - 1))/(v_1(gamma_2 - 1)+ v_2(gamma_1 - 1))`.
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Knowledge Check

  • The value of gamma=C_p/C_v for a gaseous mixture consisting of 2.0 moles of oxygen and 3.0 moles of helium. The gases are assumed to be ideal.

    A
    `7/5`
    B
    `29/19`
    C
    `23/19`
    D
    `25/19`

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