Question 5.1: a) Evaluate the speed of sound in a mixture of two ideal gas...
a) Evaluate the speed of sound in a mixture of two ideal gases of masses m_1 and m_2 occupying the unit volume. Their molar masses are M_1 and M2 and their adiabatic constants are γ_1 and γ_2 . b) Calculate the speed of sound at 20°C in air (21% of O_2 and 79% of N_2 in volume). M_O = 32 and M_N = 28.
a) The density of this mixture is m_v=(m_1+m_2). The unit volume contains n_1=m_1/M_1 moles of the first gas and n_2=m_2/M_2 moles of the second. Using the equation pV = nRT for each one, we obtain the partial pressures P_1=m_1RT/M_1 \ \text {and} \ P_2=m_2RT/M_2 . According to [5.22], if a volume V of this gas varies by δV, the variation of the total pressure is
\delta P=-B(\delta v/v) [5.22]
\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 .Thus, the global bulk modulus is B=RT(\gamma _1m_1/M_1+\gamma _2m_2/M_2) \ \text {and} \\ v_s=\sqrt{B/m_v} =\sqrt{RT(\gamma _1m_1/M_1+\gamma _2m_2/M_2)/(m_1+m_2)} .
b) The unit volume of air contains 0.21 m³ of oxygen and 0.79 m³ of nitrogen. Their masses are (in kg/m³):
m_O=0.032\times (0.21/0.0224)=0.300 \ \text {and} \ m_N=0.028\times (0.79/0.0224)=0.988.As \gamma _O=\gamma _N=1.4 , the speed of sound in air (STP) is
v_s=\left[RT(\gamma _Om_O/M_O+\gamma _Nm_N/M_N)/(m_O+m_N)\right]^½=344 \ m/s .