# Question #9802d

##### 1 Answer

#### Explanation:

Your starting point here will be the **fundamental thermodynamic relation**, which establishes a relationship between *infinitesimal changes* in **internal energy**, **entropy**, and **volume** for a closed system at thermal equilibrium

#color(blue)( |bar(ul( color(white)(a/a)dU = TdS - PdVcolor(white)(a/a)))|)" "# , where

**absolute temperature** of the gas

Now, an **isothermal expansion** is characterized by the fact that the temperature of the gas **remains unchanged**. For an ideal gas that undergoes an isothermal expansion, the fact fact that temperature is held constant implies that you have

#dU = 0#

This means that you can say

#0 = TdS - PdV implies TdS = PdV#

This can be rearranged to give

#dS = P/TdV#

Now, assuming that we're going from an initial state *change in entropy*, **integrating** the above equation

#[dS = P/TdV] -> int_color(purple)(1)^color(purple)(2)#

#DeltaS = int_color(purple)(1)^color(purple)(2) P/T dV#

Since you're dealing with an *ideal gas*, you can use the **ideal gas law** equation

#color(blue)(|bar(ul(color(white)(a/a)PV = nRTcolor(white)(a/a)))|)#

to say that

#P/T = (nR)/V = overbrace(nR)^(color(red)("constant")) * 1/V#

This means that you have

#DeltaS = nR * int_color(purple)(1)^color(purple)(2) 1/V dV#

Finally, this will get you

#DeltaS = nR * ln(V_2/V_1)#

Here

*number of moles* of gas present in the sample.

*universal gas constant*, equal to

Plug in your values to get

#DeltaS = 0.13 color(red)(cancel(color(black)("moles"))) * "8.314 J"color(red)(cancel(color(black)("mol"^(-1))))"K"^(-1) * ln((19 color(red)(cancel(color(black)("L"))))/(11color(red)(cancel(color(black)("L")))))#

#DeltaS = color(green)(|bar(ul("0.59 J K"^(-1)))|)#

*Now, does a positive change in entropy make sense?*

Notice that when **increasing** the volume of a gas *while keeping its temperature constant* will result in an increase in entropy.

That happens because the **same number of molecules** of gas are now floating around in a **larger volume**, which implies that the *randomness* and *disorder* of the system have increased.