Q 5.56 Prove that the entropy of mixin... [FREE SOLUTION]

91影视

An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 5: Q 5.56 (page 191)

Prove that the entropy of mixing of an ideal mixture has an infinite slope, when plotted vs. x, at x = 0 and x= 1.

Short Answer

Expert verified

Therefore, it is proved that the entropy of mixing of an ideal mixture has an ideal slope.

Step by step solution

Given information

The entropy of mixing of an ideal mixture has an infinite slope, when plotted vs. x, at x = 0 and x= 1.

Concept

When two ideal gases are allowed to mix at the same pressure and temperature and the total volume remains unaltered, the change in entropy is:

$螖 S_{mix} = - n R (x \ln (x) + (1 - x) \ln (1 - x))$

Explanation

Take the derivative of $螖 S_{m i x}$ to show that the slope of the entropy with respect to x is zero at x = 0 and x = 1.

$\frac{d}{d x} 螖 S_{m i x} = - n R (x 脳 \frac{1}{x} + \ln (x) - (1 - x) 脳 \frac{1}{1 - x} - \ln (1 - x)) \frac{d}{d x} 螖 S_{m i x} = n R (\ln (1 - x) - \ln (x)) \frac{d}{d x} 螖 S_{m i x} = n R \ln (\frac{1 - x}{x})$

At x=0, we have

${\frac{d}{d x} 螖 S_{m i x}|}_{x = 0} = n R \ln (\frac{1}{0}) = n R \ln (鈭�) = 鈭� {\frac{d}{d x} 螖 S_{m i x}|}_{x = 0} = 鈭�$

At x=1, we have

${\frac{d}{d x} 螖 S_{m i x}|}_{x = 1} = n R \ln (\frac{0}{1}) = n R \ln (0) = 鈭� {\frac{d}{d x} 螖 S_{m i x}|}_{x = 1} = 鈭�$

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91影视

Prove that the entropy of mixing of an ideal mixture has an infinite slope, when plotted vs. x, at x = 0 and x= 1.

Short Answer

Step by step solution

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Concept

Explanation

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