Deviations from the Ideal-Gas Equation. For carbon dioxide gas (CO2), the constants in the van d

Deviations from the Ideal-Gas Equation. For carbon dioxide...

Deviations from the Ideal-Gas Equation. For carbon dioxide gas $\left(\mathrm{CO}_{2}\right),$ the constants in the van der Waals equation are $a=0.364 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}$ and $b=4.27 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol} .$ (a) If 1.00 mol of $\mathrm{CO}_{2}$ gas at 350 $\mathrm{K}$ is confined to a volume of $400 \mathrm{cm}^{3},$ find the pressure of the gas using the ideal-gas equation and the van der Waals equation. (b) Which equation gives a lower pressure? Why? What is the percentage difference of the van der Waals equation result from the ideal-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of 4000 $\mathrm{cm}^{3} .$ Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideal-gas equation if $n / V$ is small.

Deviations from the Ideal-Gas Equation. For carbon dioxide gas $\left(\mathrm{CO}_{2}\right),$ the constants in the van der Waals equation are $a=0.364 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}$ and $b=4.27 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol} .$ (a) If 1.00
mol of $\mathrm{CO}_{2}$ gas at 350 $\mathrm{K}$ is confined to a volume of $400 \mathrm{cm}^{3},$ find the pressure of the gas using the ideal-gas equation and the van der
Waals equation. (b) Which equation gives a lower pressure? Why? What is the percentage difference of the van der Waals equation result from the ideal-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of 4000 $\mathrm{cm}^{3} .$ Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideal-gas equation if $n / V$ is small.

Key Concepts

Deviation from Ideal Behavior

Deviation analysis involves comparing the predictions of the Ideal Gas Law with those of the van der Waals equation. This process highlights under what conditions real gases begin to differ significantly from ideal behavior, particularly when intermolecular forces and molecular volume become non-negligible.

Finite Molecular Volume

Real gas molecules occupy a finite volume, in contrast to the point particles assumed in the Ideal Gas Law. The van der Waals equation includes a correction term for this finite volume, subtracting it from the actual volume available for particle motion, thereby affecting the gas pressure.

Low Density Limit

The low density (or high volume) limit is a condition wherein the number of particles per unit volume is small. Under these conditions, the corrections due to intermolecular attractions and finite molecular volume become negligible, causing the van der Waals equation to converge to the Ideal Gas Law.

Van der Waals Equation

The van der Waals equation modifies the Ideal Gas Law by incorporating corrections for intermolecular attractions and the finite volume occupied by gas molecules. It introduces constants to account for these deviations, providing a more accurate description of real gases, especially under high pressure or low temperature conditions.

Ideal Gas Law

The Ideal Gas Law (PV = nRT) is a fundamental relation that describes the behavior of an idealized gas, assuming point particles with no intermolecular interactions. It serves as a baseline for understanding gas behavior under various conditions, particularly when gas molecules are widely separated.

Intermolecular Forces Correction

This concept refers to the attractive forces between gas molecules, which are not considered in the Ideal Gas Law. In the van der Waals equation, these forces are represented by an attraction constant, reducing the effective pressure exerted by the gas as molecules attract each other.

Transcript

00:01 Hello, so in this question we have a gas carbon dioxide and we're trying to find the pressure under two conditions the first condition is that it's an ideal gas so we use the ideal gas equation the second condition is that it's a real gas and so we use the van der waals equation so van der waals equation takes into account the intermolecular forces between the individual molecules in the gas but an ideal gas doesn't so what we do is we're going to find the pressure assuming that is an ideal gas so we use the ideal gas equation pv equals nr t p is the pressure v is the volume n is the number of moles r is a constant and t is the temperature in kelvin so when you isolate the p for the idle gas equation you get p equals nrt over v so i'm going to go ahead and do the substitution alright so we're going to do the substitution the number of moles is just one r is a constant which is 8 .315 temperature is 350 the volume given was 400 cmq so we need to convert that to cubic meters so you're going to have 400 times 10 to the minus 6.

01:51 If you put this in the calculator, you're going to get 7 .28 times 10 .6 pascal.

02:06 So this is the pressure of carbon dioxide, assuming it's an ideal gas.

02:14 For a real gas, we use the van der weals equation, given by p equals rt, all over minus, all over, v minus b sorry everything minus a over v squared again a and b are just constants so we're giving in a question so we just have to go ahead and do our substitutions we have our r which is 8 .315 one temperature is 350 of volume 400 10 is minus 6 minus the constant b i was given at 4 .2 6 4 .27.

03:12 I have to extend this a little bit.

03:16 0 .27 times 10.

03:19 That is negative 5 minus e.

03:27 Is given us 0 .364.

03:35 4 divided by v squared, which is less 400 times 10 minus 6 squared.

03:54 Let me do that right.

03:56 Minus 6 squared so if you put this in a calculator you should get something around 5 .87 times 10 esplan 6 pascal so that is the first part of the question to just find the pressure assuming it's ideal and then when it is a real gas where we use the van der waaz equation for the second part, we have been asked to determine which equation gives a lower pressure and why...

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