a) Evaluate the speed of sound in a mixture of two ideal gases of masses m1 and m2 occupying the unit volume. Their molar masses are M1 and M2 and their adiabatic constants are γ1 and γ2. b) Calculate the speed of sound at 20°C in air (21% of O2 and 79% of N2 in volume). MO = 32 and MN = 28.

Question

a) Evaluate the speed of sound in a mixture of two ideal gases of masses [latex] m_1 [/latex] and [latex] m_2 [/latex] occupying the unit volume. Their molar masses are [latex] M_1 [/latex] and [latex] M2 [/latex] and their adiabatic constants are [latex] γ_1 [/latex] and [latex] γ_2 [/latex] . b) Calculate the speed of sound at 20°C in air (21% of [latex] O_2 [/latex] and 79% of [latex] N_2 [/latex] in volume). [latex] M_O = 32 [/latex] and [latex] M_N = 28. [/latex]

Accepted Answer

a) The density of this mixture is [latex] m_v=(m_1+m_2). [/latex] The unit volume contains [latex] n_1=m_1/M_1 [/latex] moles of the first gas and [latex] n_2=m_2/M_2 [/latex] moles of the second. Using the equation pV = nRT for each one, we obtain the partial pressures [latex] P_1=m_1RT/M_1 \ ext {and} \ P_2=m_2RT/M_2 . [/latex] According to [5.22], if a volume V of this gas varies by δV, the variation of the total pressure is

[latex] \delta P=-B(\delta v/v) [/latex] [5.22]

[latex] \delta P=\delta P_1+\delta P_2=-\gamma _1P_1 \ \delta v/v-\gamma _2P_2 \ \delta v/v=-RT(\gamma _1m_1/M_1+\gamma _2m_2/M_2) \ \delta v/v . [/latex]

Thus, the global bulk modulus is [latex] B=RT(\gamma _1m_1/M_1+\gamma _2m_2/M_2) \ ext {and} \ v_s=\sqrt{B/m_v} =\sqrt{RT(\gamma _1m_1/M_1+\gamma _2m_2/M_2)/(m_1+m_2)} . [/latex]

b) The unit volume of air contains 0.21 m³ of oxygen and 0.79 m³ of nitrogen. Their masses are (in kg/m³):

[latex] m_O=0.032 imes (0.21/0.0224)=0.300 \ ext {and} \ m_N=0.028 imes (0.79/0.0224)=0.988. [/latex]

As [latex] \gamma _O=\gamma _N=1.4 [/latex] , the speed of sound in air (STP) is

[latex] v_s=\left[RT(\gamma _Om_O/M_O+\gamma _Nm_N/M_N)/(m_O+m_N) ight]^½=344 \ m/s [/latex] .

Question 5.1: a) Evaluate the speed of sound in a mixture of two ideal gas...

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