Chapter 6: Q. 6.46 (page 255)

Equations 6.92 and 6.93 for the entropy and chemical potential involve the logarithm of the quantity $\frac{V Z_{i n t}}{N v_{Q}}$ . Is this logarithm normally positive or negative? Plug in some numbers for an ordinary gas and discuss.

Short Answer

Expert verified

Yes, the logarithm is normally positive for ordinary gas as $∆ S = N k \ln 2$ is positive.

Step by step solution

Given information

We are given that,

Entropy and chemical potential involve the logarithm of the quantity $\frac{V Z_{i n t}}{N v_{Q}}$ .

Explanation

We know that entropy is given as,

$S = k N \{\ln [\frac{V}{N} {(\frac{4 πm}{3 h^{2}} \times \frac{U}{N})}^{\frac{3}{2}}] + \frac{5}{2}\}$

Here, $U$ is the energy

and $V$ is volume

and $N$ is no. of molecules

Now, if energy and no. of molecules remain fixed,

$∆ S = N k \frac{V_{f}}{V_{i}}$

Here, $V_{f}$ is the final volume

and $V_{i}$ is the initial volume

If $V_{f} = 2 V_{i}$

Then,

$∆ S = N k \ln 2$

Similarly, the relationship between chemical potential can be determined to be positive.

From here, we can interpret that the logarithm of a given quantity is normally positive as both the entropy and chemical potential is positive.

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Equations 6.92 and 6.93 for the entropy and chemical potential involve the logarithm of the quantity $\frac{V Z_{i n t}}{N v_{Q}}$ . Is this logarithm normally positive or negative? Plug in some numbers for an ordinary gas and discuss.

Short Answer

Step by step solution

Given information

Explanation

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Equations 6.92 and 6.93 for the entropy and chemical potential involve the logarithm of the quantity VZintNvQ. Is this logarithm normally positive or negative? Plug in some numbers for an ordinary gas and discuss.

Short Answer

Step by step solution

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Explanation

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Equations 6.92 and 6.93 for the entropy and chemical potential involve the logarithm of the quantity $\frac{V Z_{i n t}}{N v_{Q}}$ . Is this logarithm normally positive or negative? Plug in some numbers for an ordinary gas and discuss.