Question

A sample of gas occupies \(14.3 \ell\) at \(19^{\circ} \mathrm{C}\) and \(790 \mathrm{~mm} \mathrm{Hg}\). Determine the number of moles of gas present, what volume will this same amount of gas occupy at \(190^{\circ} \mathrm{C}\) and \(79.0 \mathrm{~mm}\) \(\mathrm{Hg}\) ?

Short answer

The number of moles of the gas in the sample is approximately 0.604 mol, and the volume of the gas under the new conditions is approximately 225.48 L.

Step-by-step solution

Write down the Ideal Gas Law

The Ideal Gas Law is given by: \(PV = nRT\) Where: P = pressure V = volume n = number of moles of the gas R = universal gas constant (R = 0.0821 L·atm/(mol·K)) T = temperature in Kelvin

Convert given values to appropriate units

Before we proceed in using the Ideal Gas Law to find the number of moles of the gas in the given sample, we must convert the given values to the appropriate units. The temperature should be in Kelvin and the pressure should be in atm. - Convert temperature from Celsius to Kelvin: T(K) = 19°C + 273.15 = 292.15 K - Convert pressure from mmHg to atm: P(atm) = 790 mmHg × (1 atm / 760 mmHg) = 1.039 atm Now we have: V = 14.3 L P = 1.039 atm T = 292.15 K

Calculate the number of moles using the Ideal Gas Law

Rearrange the Ideal Gas Law formula to solve for n, the number of moles of the gas: \(n = \frac{PV}{RT}\) Plug in the known values: \(n = \frac{(1.039 \,\text{atm})(14.3\,\text{L})}{(0.0821\, \frac{\text{L}\cdot\text{atm}}{\text{mol}\cdot\text{K}})(292.15\, \text{K})}\) Calculate the result: \(n \approx 0.604 \,\text{mol}\) The number of moles of the gas in the sample is approximately 0.604 mol.

Write down the Combined Gas Law

We will use the Combined Gas Law to determine the new volume of the gas under different temperature and pressure conditions. The Combined Gas Law is given by: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\)

Convert the new conditions to appropriate units

Convert the new temperature and pressure to Kelvin and atm, respectively: - New Temperature T2 = 190°C + 273.15 = 463.15 K - New Pressure P2 = 79.0 mmHg × (1 atm / 760 mmHg) = 0.1039 atm Now we have: P1 = 1.039 atm V1 = 14.3 L T1 = 292.15 K P2 = 0.1039 atm T2 = 463.15 K

Substitute values into the Combined Gas Law to find the new volume

Rearrange the Combined Gas Law to solve for V2, the new volume: \(V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}\) Plug in the known values: \(V_2 = \frac{(1.039\,\text{atm})(14.3\,\text{L})(463.15\, \text{K})}{(0.1039\, \text{atm})(292.15\,\text{K})}\) Calculate the result: \(V_2 \approx 225.48\, \text{L}\) The volume of the gas under the new conditions is approximately 225.48 L.

Key concepts

Combined Gas Law

The Combined Gas Law is a fundamental concept in chemistry that combines three gas laws: Boyle's Law, Charles' Law, and Gay-Lussac's Law. It allows us to predict how a gas will change under varying pressure (P), volume (V), and temperature (T) conditions without needing to know the amount of gas present.

The formula for the Combined Gas Law is expressed as:
\[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]
This formula states that the ratio of pressure and volume to temperature for any gas is a constant for an isolated system as long as the amount of gas (number of moles) doesn't change.

To apply it practically, like in the given exercise, we simply plug in known values for initial pressure, volume, temperature (labeled with subscript 1) and final pressure, temperature (labeled with subscript 2) to find the unknown final volume (V2). It effectively illustrates the inversely proportional relationship between pressure and volume, and the directly proportional relationship between volume and temperature.

Molar Volume

Molar Volume is the volume that one mole of any gas occupies at a given temperature and pressure. It is a cornerstone concept in gas stoichiometry and the Ideal Gas Law. Under standard temperature and pressure (0°C and 1 atm), one mole of any ideal gas occupies 22.4 liters, known as the standard molar volume.

While the molar volume can change with different conditions, this is where the Ideal Gas Law comes in handy. The Ideal Gas Law, represented by \(PV = nRT\), allows us to find the molar volume at conditions other than standard temperature and pressure by rearranging this equation to \(V = \frac{nRT}{P}\).

In the exercise provided, by finding the number of moles in 14.3 liters of gas using the Ideal Gas Law, we can subsequently determine the molar volume under the new conditions. It's important to note that the molar volume will differ depending on the specific temperature and pressure conditions of the system.

Gas Stoichiometry

Gas Stoichiometry involves the calculation of reactants and products in chemical reactions involving gases. It uses the relationships defined in the Ideal Gas Law and often involves molar volume as well. Understanding stoichiometry is imperative for chemists when they need to calculate the amounts of substances produced or used in a chemical reaction.

One key aspect of gas stoichiometry is the use of molar ratios from the balanced chemical equation. These ratios, combined with the knowledge of the molar volume of a gas, enable the conversion between volume and moles, which is an essential step in stoichiometric calculations.

Through the solution to the provided exercise, we can observe gas stoichiometry; first by using the Ideal Gas Law to find the number of moles in a given volume and then applying the Combined Gas Law to predict how these moles react when subjected to different conditions of temperature and pressure. This predictive nature of gas stoichiometry makes it invaluable for planning chemical processes and understanding gas behaviors.

Always remember, for accurate stoichiometric calculations in gaseous systems, it is important to account for all state variables of the gas in question and ensure that units are consistent across all variables.