Question

What pressure (in atm) is exerted by \(1.51 \times 10^{23}\) oxygen molecules at \(25^{\circ} \mathrm{C}\) in a 5.00 -L container?

Short answer

The pressure exerted is approximately 1.232 atm.

Step-by-step solution

Convert Molecules to Moles

To find the moles of oxygen, divide the number of molecules by Avogadro's number. \[ ext{Moles of O}_2 = \frac{1.51 imes 10^{23}}{6.022 imes 10^{23} ext{ mol}^{-1}} \approx 0.251 ext{ mol} \]

Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin.\[ T( ext{K}) = 25^{ ext{C}} + 273.15 = 298.15 ext{ K} \]

Use the Ideal Gas Law

The ideal gas law is \( PV = nRT \). We need to solve for the pressure \( P \). Substitute in the values where \( n = 0.251 \text{ mol}, R = 0.0821 \text{ L} \, ext{atm} \, ext{K}^{-1} \, ext{mol}^{-1}, T = 298.15 \text{ K}, V = 5.00 \text{ L} \).\[ P = \frac{nRT}{V} = \frac{(0.251)(0.0821)(298.15)}{5.00} \approx 1.232 \text{ atm} \]

Key concepts

Moles and Molecules Conversion

Understanding the relationship between moles and molecules is key when dealing with chemical equations and problems. In chemistry, the mole is a standard unit to express the amount of a substance. One mole contains exactly Avogadro's number of molecules, which is approximately \(6.022 \times 10^{23}\) molecules. This large number allows chemists to count entities at a molecular level by weighing out grams. To convert molecules to moles, you divide the number of molecules by Avogadro's number. For example, given \(1.51 \times 10^{23}\) molecules, you determine the moles by:\[ \text{Moles} = \frac{1.51 \times 10^{23}}{6.022 \times 10^{23}} \approx 0.251 \text{ mol} \]This conversion is crucial as it allows the use of the Ideal Gas Law, where quantities need to be in terms of moles rather than molecules.

Temperature Conversion

Converting temperature from Celsius to Kelvin is a necessary step when working with gas laws. This conversion is particularly important because gas law equations require the absolute temperature scale, which Kelvin provides. The Kelvin scale starts at absolute zero, the theoretical temperature where particles have minimum thermal motion.The conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature:\[T(\text{K}) = T(\degree C) + 273.15\]For example, converting 25°C to Kelvin would be:\[T(\text{K}) = 25 + 273.15 = 298.15 \text{ K}\]Performing this conversion ensures that pressure and volume calculations are accurate, as the Ideal Gas Law depends on temperature input in Kelvin.

Pressure Calculation

To find the pressure exerted by a gas using the Ideal Gas Law, one must understand how to manipulate the equation: \[ PV = nRT \]Where:

• \(P\) represents the pressure of the gas
• \(V\) is the volume which, in this instance, is 5.00 L
• \(n\) is the number of moles, which we have calculated as 0.251 mol
• \(R\) is the Ideal Gas Constant (0.0821 L atm K^-1 mol^-1)
• \(T\) is the temperature in Kelvin (298.15 K)

The goal is to find \(P\), thus rearranging the equation gives:\[ P = \frac{nRT}{V} \]Substituting the determined values:\[ P = \frac{(0.251)(0.0821)(298.15)}{5.00} \approx 1.232 \text{ atm} \]This calculated pressure, expressed in atm (atmospheres), quantifies the force exerted by the gas within the container. Understanding pressure calculation is vital in numerous scientific and industrial applications.