The molar specific heat at constant pressure for a certain...

The molar specific heat at constant pressure for a certain gas is given by $C_{p}=a+b T+c T^{2},$ where $a=33.6 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}$ $\mathrm{b}=2.93 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2},$ and $c=2.13 \times 10^{-5} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{3} .$ Find the entropy change when 2.00 moles of this gas are heated from $20.0^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$

The molar specific heat at constant pressure for a certain gas is given by $C_{p}=a+b T+c T^{2},$ where $a=33.6 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}$
$\mathrm{b}=2.93 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2},$ and $c=2.13 \times 10^{-5} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{3} .$ Find
the entropy change when 2.00 moles of this gas are heated from $20.0^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$

Key Concepts

Temperature Conversion to Kelvin

In thermodynamic calculations, it is essential to use the absolute temperature scale (Kelvin) to avoid inconsistencies and to correctly apply the laws of thermodynamics. Temperature conversion ensures that the integration of thermodynamic quantities is done accurately.

Integration of Temperature-Dependent Specific Heat

When the specific heat capacity is a function of temperature, as in the given case, the entropy change must be determined by integrating the expression C?(T)/T with respect to temperature. This integration accounts for the variations in heat capacity over the temperature range of the process.

Entropy

Entropy is a measure of the disorder or randomness in a system, and in thermodynamics it is a state function that quantifies the dispersal of energy in a process. Changes in entropy can be calculated for processes by integrating the heat transfer divided by the temperature, particularly in reversible transformations.

Specific Heat Capacity at Constant Pressure

The specific heat capacity at constant pressure (C?) quantifies the amount of heat required to raise the temperature of one mole of a substance by one Kelvin while maintaining constant pressure. It is a key property in calculating the heat transfer and subsequent entropy changes in thermal processes.

Transcript

00:01 So in this example, we're trying to find the entropy change of a gas going from 20 degrees celsius to 200 degrees celsius.

00:07 And we're given a specific heat equation given by cp with the constants a, b, and c.

00:14 And it's two moles of gas.

00:16 So we write out the basic definition of entropy.

00:21 We're going to start out with delta s is equal to the integral from t1 to t2 of dq over t.

00:35 So we can just write this over here.

00:38 That t1 is going to be 20 degrees celsius.

00:42 So if we put that in kelvin, we just add 273, so it'll be 293 kelvin.

00:50 And t2 is 200 degrees celsius.

00:56 So that'll be 473 kelvin.

01:02 All right.

01:03 So we'll just put those in at the end.

01:05 And here we also know that dq is equal to ncp.

01:13 D t and we're told that it is two moles of gas so n is equal to two cp is equal to this equation over here that we're given and then d t will just be used in the integration process so we can rewrite this that's s is equal to the integral t1 to t2 and again we'll just put those values in at the end it'll just make our lives a little bit easier so then we've got n which is we'll just keep that as n2 and a plus bt plus c t squared, all divided by t, and this is integrated with respect to temperature.

02:03 So we can take n out because it's a constant, and just rewrite this again, and then we're going to have to integrate three different times...

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