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Chapter 10: Problem 17

Which gas, \(\mathrm{CO}\) or \(\mathrm{N}_{2},\) is expected to behave more ideally at 10 \(\mathrm{atm}\) and \(-50^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
At 10 atm and -50°C, the molar volumes of both CO and N2 are approximately 1.825 L/mol, and both have a molecular weight of 28 g/mol. However, N2 has a smaller molecular size, leading to less significant intermolecular forces and more ideal behavior. Therefore, N2 is expected to behave more ideally under these conditions than CO.

Step by step solution

01

Convert temperature to Kelvin

Convert the given temperature from Celsius to Kelvin, using the formula, \(K = ^{\circ}C + 273.15\). \(-50^{\circ}C + 273.15 = 223.15 K\) So, the temperature is 223.15 K.
02

Use the Ideal Gas Law

The Ideal Gas Law is given by the equation: \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. We know that pressure (P) = 10 atm, R is the ideal gas constant (∼0.0821 L atm / K mol), and T = 223.15 K. Rearrange the Ideal Gas Law to find the volume occupied by 1 mole of each gas: \(V_m = \frac{nRT}{P}\) For 1 mole of gas (n = 1), we can find the molar volume as: \(V_m = \frac{RT}{P}\)
03

Calculate the molar volumes of CO and N2

We will now calculate the molar volume for each gas using the formula we derived in step 2: \(V_m = \frac{RT}{P}\) For CO: \(V_m = \frac{(0.0821 \hspace{2} L \hspace{2} atm \hspace{2} mol^{-1} \hspace{2} K^{-1})(223.15 \ hsapce{2} K)}{10 \hspace{2} atm} \) \(V_m \approx 1.825 L/mol\) For N2: \(V_m = \frac{(0.0821 \hspace{2} L \hspace{2} atm \hspace{2} mol^{-1} \hspace{2} K^{-1})(223.15 \ hsapce{2} K)}{10 \hspace{2} atm} \) \(V_m \approx 1.825 L/mol\) Since both gases have the same molar volume, we need to compare the molecular weight and size of each gas to determine which one would behave more ideally.
04

Compare the molecular weight and size

The molecular weight of CO is 28 g/mol, and the molecular weight of N2 is also 28 g/mol. Both gases have the same molecular weight. However, the size of the N2 molecule is smaller than that of CO. The smaller size means less significant intermolecular forces, leading to more ideal behavior.
05

Conclusion

Even though CO and N2 have the same molar volume and molecular weight, the smaller size of the N2 molecule allows it to behave more ideally under the given conditions. So, we can expect N2 to behave more ideally than CO at 10 atm and -50°C.

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