Problem 29 - Chemistry Matter and Change

Question

If the pressure exerted by a gas at \(25^{\circ} \mathrm{C}\) in a volume of 0.044 L is 3.81 atm, how many moles of gas are present?

Step-by-Step Solution

Answer

There are approximately 0.0068 moles of gas present.

Step 1: Understanding the Problem

We are given the pressure, volume, and temperature of a gas, and we need to find the number of moles. This is a typical problem where the Ideal Gas Law can be used.

Step 2: Write the Ideal Gas Law Formula

The Ideal Gas Law is given by the formula \( PV = nRT \), 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 in Kelvin.

Step 3: Convert Temperature to Kelvin

The temperature given is \(25^{\circ}C\). To convert to Kelvin, add 273.15. So, \( T = 25 + 273.15 = 298.15 \, K \).

Step 4: Use the Ideal Gas Constant

The gas constant \( R \) is typically \(0.0821 \, \text{L atm} \, \text{mol}^{-1} \, \text{K}^{-1}\). We will use this value for \( R \).

Step 5: Substitute into the Ideal Gas Law

Now substitute the known values into the Ideal Gas Law: \( P = 3.81 \, \text{atm} \), \( V = 0.044 \, \text{L} \), \( T = 298.15 \, \text{K} \), and \( R = 0.0821 \, \text{L atm} \, \text{mol}^{-1} \, \text{K}^{-1}\). We are solving for \( n \), so rearrange the formula: \( n = \frac{PV}{RT} \).

Step 6: Calculate the Number of Moles

Substitute the values into the formula: \( n = \frac{3.81 \, \text{atm} \times 0.044 \, \text{L}}{0.0821 \, \text{L atm} \, \text{mol}^{-1} \, \text{K}^{-1} \times 298.15 \, \text{K}} \). Calculating this gives \( n \approx 0.0068 \, \text{mol} \).

Key Concepts

Gas Pressure

Gas pressure is an important concept when dealing with gases. It refers to the force that the gas particles exert on the walls of their container.

• The amount of pressure a gas exerts depends on various factors like the volume of the container and the temperature of the gas.
• Pressure is typically measured using units such as atmospheres (atm), pascals (Pa), or millimeters of mercury (mmHg).

In our original exercise, the pressure of the gas is given as 3.81 atm. This is one of the key variables in the Ideal Gas Law equation, helping to relate the various physical characteristics of gases. When calculating reactions or performing conversions at different conditions, understanding gas pressure and its measurement is crucial.

Temperature Conversion to Kelvin

When dealing with gas calculations, it's essential to work with temperature in Kelvin. The Kelvin scale starts at absolute zero, the point where atomic movement ceases, making it ideal for scientific calculations.

• To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.
• This conversion is important because Kelvin provides a direct correlation with the kinetic energy of particles, aligning perfectly with the gas laws.

For example, in our problem, the given temperature is 25°C. By adding 273.15, we convert this to 298.15 K. By using Kelvin, we ensure that our Ideal Gas Law calculations are accurate.

Moles Calculation

In gas chemistry, the term 'moles' refers to the number of atoms or molecules in a given amount of substance. This is crucial when applying the Ideal Gas Law.

• The Ideal Gas Law allows us to solve for moles with the equation: \( n = \frac{PV}{RT} \).
• It's important to have all units consistent with the gas constant used (which is discussed in the next section) for correct calculations.

From our exercise, substituting the given values (pressure, volume, gas constant, and temperature) into the formula, we find there are approximately 0.0068 moles of the gas present. Understanding moles helps you quantify the amount of substance involved in reactions and calculations.

Gas Constant

The gas constant, often symbolized by \( R \), is a critical component of the Ideal Gas Law. It serves as a bridge linking various units in the equation, such as pressure, volume, temperature, and moles.

• The value of \( R \) depends on the units of pressure, volume, and temperature used in the equation. In our exercise, \( R \) is 0.0821 L atm mol\(^{-1}\) K\(^{-1}\).
• This specific form of \( R \) is applicable when pressure is in atmospheres, volume in liters, and temperature in Kelvin. Other forms exist for different sets of units.

Having the correct value of \( R \) is essential to ensure accurate calculations when using the Ideal Gas Law. By using the right constant, any discrepancies in measurement units are effectively managed, allowing for precise results.