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How does one solve the following differential equation? This is a reduced mass balance equation for a cylinder containing a gas mixture of two gases (here represented by the numbers 1 and 2)):

$$\frac{p_1C_{V_1}^*}{RT_1}\frac{\mathrm dT_1}{\mathrm dt}+\frac{p_2C_{V_2}^*}{RT_2}\frac{\mathrm dT_2}{\mathrm dt}=\frac{\mathrm d}{\mathrm dt}\left(\frac{p_1}{T_1}\right)T_1+\frac{\mathrm d}{\mathrm dt}\left(\frac{p_2}{T_2}\right)T_2$$

For one gas it's easy:

$$\frac{pC_V^*}{RT}\frac{\mathrm dT}{\mathrm dt}=\frac{\mathrm d}{\mathrm dt}\left(\frac pT\right)T$$

$$\frac{C_V^*}{RT}\frac{\mathrm dT}{\mathrm dt}=\frac{\mathrm d}{\mathrm dt}\left(\frac pT\right)\frac Tp$$

$$\frac{C_V^*}{R}\frac{\mathrm d\ln(T)}{\mathrm dt}=\frac{\mathrm d}{\mathrm dt}\left(\ln\left(\frac pT\right)\right)$$

After integrating between the initial and final phase, we get:

$$\left(\frac{T_\text{final}}{T_\text{initial}}\right)^{\frac{C_V^*}R}=\left(\frac{p_\text{final}}{p_\text{initial}}\right)\left(\frac{T_\text{initial}}{T_\text{final}}\right)$$

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  • $\begingroup$ What does $T_1$ and $T_2$ represent? In most practical situations, it would be impossible for each gas in a mixture of gases to have separate temperatures. Also, does $P_1$ represent the partial pressure of gas 1? Or the total pressure. (If the latter, what is $P_2$?) $\endgroup$ – Curt F. Oct 20 '15 at 19:40
  • $\begingroup$ Yes, some gases tend to migrate to warmer regions, others to colder regions. Anyway, for simplicity's sake, you could say that $T_{1}=T_{2}$. $P_{1}$ and $P_{2}$ are the partial pressures of gas 1 and 2 resp (n.b. these aren't equal) and $C_{V_{1}}^{*}$ and $C_{V_{2}}^{*}$ are the specific heat capacities of each gas at constant volume. $\endgroup$ – stanley Oct 21 '15 at 5:10
  • $\begingroup$ If $T_1=T_2$, then $dT_1 = dT_2$ and you could considerably simplify the equation. Also if the gases aren't reacting then can't you just calculate an appropriately averaged $C_v$ and then use the single-component gas equation? $\endgroup$ – Curt F. Oct 21 '15 at 5:24
  • $\begingroup$ yes, but in that case you merge everything and you get the solution I wrote above and I would like to consider the thermal gradient within the cylinder containing the gas in the equation $\endgroup$ – stanley Oct 22 '15 at 15:52
  • $\begingroup$ If you are interested in modeling thermal gradients, why isn't there any spatial variable in your question? Could you edit your question to expand on exactly what -- chemically -- is confusing you about the equation? I'm not sure but from the comments it sounds like you are interested in far more than just solving the equation you wrote. What does it represent? How did you derive it? Do you think it has anything to do with spatial gradients? If you add more info to the question you'll get a better answer. $\endgroup$ – Curt F. Oct 22 '15 at 16:34

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