202109160857 Exercises 5.1.A (Q17)

Integrate both sides of the equation

$\displaystyle{\frac{\mathrm{d}P}{P}+\frac{\mathrm{d}M}{M}=\frac{\mathrm{d}T}{2T}}$

to obtain the ideal gas continuity relation:

$\displaystyle{\frac{PM}{\sqrt{T}}=\textrm{constant}}$ .

Attempts.

$\begin{aligned} \int\frac{\mathrm{d}P}{P} + \int\frac{\mathrm{d}M}{M} & = \int\frac{\mathrm{d}T}{2T} \\ \ln |P| + \ln |M| - \frac{1}{2}\ln |T| & = C \\ \ln \bigg|\frac{PM}{T^{1/2}}\bigg| & = C \\ \frac{PM}{\sqrt{T}} & = \textrm{Const.} \\ \end{aligned}$

M	T	W	T	F	S	S
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30

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