Question

Given each of the following sets of values for an ideal gas, calculate the unknown quantity. a. \(P=1.01\) atm; \(V=? ; n=0.00831\) mol \(; T=25^{\circ} \mathrm{C}\) b. \(P=? \mathrm{atm} ; V=602 \mathrm{mL} ; n=8.01 \times 10^{-3} \mathrm{mol} ; T=\) \(310 \mathrm{K}\) c. \(P=0.998 \mathrm{atm} ; V=629 \mathrm{mL} ; n=? \mathrm{mol} ; T=35^{\circ} \mathrm{C}\)

Short answer

a. The volume is approximately \(0.0206\, \mathrm{L}\). b. The pressure is approximately \(1.345\, \mathrm{atm}\). c. The number of moles is approximately \(0.00255\, \mathrm{mol}\).

Step-by-step solution

a. Finding Volume:

First, convert the temperature from Celsius to Kelvin: \(T = 25 + 273.15 = 298.15 \mathrm{K}\) Now, rearrange the Ideal Gas Law equation to find the volume: \(V = \frac{nRT}{P}\) Substitute the given values and the ideal gas constant, R = 0.08206 (L·atm)/(K·mol): \(V = \frac{(0.00831\,\mathrm{mol})(0.08206\,\frac{\mathrm{L·atm}}{\mathrm{K·mol}})(298.15\, \mathrm{K})}{1.01\, \mathrm{atm}}\) Now, calculate the volume: \( V \approx 0.0206\, \mathrm{L} \)

b. Finding Pressure:

In this case, the temperature is already in Kelvin. Convert the volume from mL to L: \(V = \frac{602\, \mathrm{mL}}{1000\, \mathrm{mL/L}} = 0.602\, \mathrm{L}\) Rearrange the Ideal Gas Law equation to find the pressure: \(P = \frac{nRT}{V}\) Substitute the given values and the ideal gas constant, R = 0.08206 (L·atm)/(K·mol): \(P = \frac{(8.01 \times 10^{-3} \mathrm{mol})(0.08206\, \frac{\mathrm{L·atm}}{\mathrm{K·mol}})(310\,\mathrm{K})}{0.602\,\mathrm{L}}\) Now, calculate the pressure: \( P \approx 1.345\, \mathrm{atm} \)

c. Finding Moles:

First, convert the temperature from Celsius to Kelvin: \(T = 35 + 273.15 = 308.15\,\mathrm{K}\) Next, convert the pressure from atm to kPa: \(P = 0.998\,\mathrm{atm} \times \frac{101.325\,\mathrm{kPa}}{1\,\mathrm{atm}} \approx 101.127\,\mathrm{kPa}\) Now, convert the volume from mL to L: \(V = \frac{629\,\mathrm{mL}}{1000\,\mathrm{mL/L}} = 0.629\,\mathrm{L}\) Rearrange the Ideal Gas Law equation to find the moles: \(n = \frac{PV}{RT}\) Substitute the given values and the ideal gas constant, R = 8.314 (L·kPa)/(K·mol): \(n = \frac{(101.127\,\mathrm{kPa})(0.629\,\mathrm{L})}{(8.314\,\frac{\mathrm{L·kPa}}{\mathrm{K·mol}})(308.15\,\mathrm{K})}\) Now, calculate the number of moles: \( n \approx 0.00255\,\mathrm{mol} \)

Key concepts

Pressure Calculation

In the world of ideal gases, understanding how to calculate pressure is essential. Pressure, in terms of gases, refers to the force exerted by gas molecules colliding with the walls of their container. It's a crucial variable in the Ideal Gas Law, which is expressed as:\[ PV = nRT \]where \(P\) stands for pressure, \(V\) is volume, \(n\) represents moles of gas, \(R\) is the ideal gas constant, and \(T\) is the absolute temperature in Kelvin.
To find pressure, the equation is rearranged as:\[ P = \frac{nRT}{V} \]Here's what you need to remember:

• Ensure the volume \(V\) is in liters (L), not milliliters (mL). Always convert if necessary by dividing by 1000.
• The temperature \(T\) must be in Kelvin, so convert from Celsius by adding 273.15.
• Use the ideal gas constant \(R = 0.08206\) (L·atm)/(K·mol) when dealing with atmospheres (atm) and liters (L).

Understanding these conversions and calculations will make it easier to determine pressure in various gas-related problems.

Volume Calculation

Volume is a measure of the space occupied by a gas. In the Ideal Gas Law, volume is pivotal and must be carefully calculated when other variables are known. To calculate the volume of an ideal gas, the Ideal Gas Law is rearranged as follows:\[ V = \frac{nRT}{P} \]Here's how to approach volume calculation:

• Convert temperature from Celsius to Kelvin using the formula \(T = T_{\text{Celsius}} + 273.15\).
• If given pressure in atmospheres (atm), use \(R = 0.08206\) (L·atm)/(K·mol) and ensure the pressure, \(P\), is in atm.
• Moles \(n\) must be in moles (mol), and the calculated volume \(V\) will be in liters.

These steps help ensure accuracy in determining the volume, especially in conditions where the gaseous behavior can be approximated as ideal.

Mole Calculation

Calculating the number of moles of a gas is another critical aspect of applying the Ideal Gas Law. Moles tell us how much of a substance is present in a system. When we use the Ideal Gas Law to solve for moles, it looks like this:\[ n = \frac{PV}{RT} \]Key points for mole calculation include:

• Pressure \(P\) should be in kilopascal (kPa) if using \(R = 8.314\) (L·kPa)/(K·mol). Convert atmospheres (atm) to kPa by multiplying by 101.325 if needed.
• Volume \(V\) should be in liters (L). Convert mL to L by dividing by 1000.
• Temperature \(T\) must always be in Kelvin. Convert from Celsius by adding 273.15.

Understanding and accurately performing these conversions and calculations ensures precise results when determining the number of moles in a gas sample.