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Chapter 5: Problem 47

A 2.10-L vessel contains 4.65 g of a gas at 1.00 atm and \(27.0^{\circ} \mathrm{C}\). (a) Calculate the density of the gas in grams per liter. (b) What is the molar mass of the gas?

Short Answer

Expert verified
The density of the gas is 2.21 g/L and the molar mass of the gas is 54.7 g/mol.

Step by step solution

01

Convert Temperature from Celsius to Kelvin

Using the conversion \(K = °C + 273.15\), the provided temperature value of \(27.0 °C\) becomes \(27.0 °C + 273.15 = 300.15 K\).
02

Find Gas Density

The gas density formula is \(D = \frac{m}{V}\), where \(D\) is density, \(m\) is the mass of gas (4.65 g), and \(V\) is the volume of gas (2.10 L). So, \(D = \frac{4.65 g}{2.10 L} = 2.21 g/L\).
03

Determine Molar Mass using the Ideal Gas Law

The ideal gas law formula is \(PV = nRT\), where \(P\) is pressure (1.00 atm), \(V\) is volume (2.10 L), \(n\) is number of moles, \(R\) is gas constant (0.0821 \(L \cdot atm / (K \cdot mol\))), and \(T\) is temperature (300.15 K). To calculate the number of moles \(n\), it's rearranged to \(n = \frac{PV}{RT}\). Substituting, \(n = \frac{1.00 atm \cdot 2.10 L}{0.0821 L \cdot atm / (K \cdot mol) \cdot 300.15 K} = 0.085 mol\). The formula for molar mass is \(MM = \frac{m}{n}\), where \(m\) is the mass of gas (4.65 g) and \(n\) is number of moles (0.085 mol). Substituting the values, \(MM = \frac{4.65 g}{0.085 mol} = 54.7 g/mol\).

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