Problem 69 - Chemistry Matter and Change

Question

Calculate the number of moles of \(\mathrm{O}^{2}\) gas held in a sealed, \(2.00-\mathrm{L}\) tank at 3.50 atm and \(25.0^{\circ} \mathrm{C}\) . How many moles would be in the tank if the temperature was raised to \(49.0^{\circ} \mathrm{C}\) and the pressure remained constant?

Step-by-Step Solution

Answer

Initial moles: 0.286; Moles at 49°C: 0.309.

Step 1: Utilize the Ideal Gas Law

Start by using the Ideal Gas Law formula, which states \( PV = nRT \). This will help us determine the moles of gas present at the initial conditions. Here, \(P\) is the pressure (3.50 atm), \(V\) is the volume (2.00 L), \(n\) is the number of moles, \(R\) is the ideal gas constant (0.0821 L atm/mol K), and \(T\) is the temperature in Kelvin.

Step 2: Convert the Temperature

Convert the initial temperature from Celsius to Kelvin using the formula \( T(K) = T(°C) + 273.15 \). For the initial temperature of \(25.0^{\circ}C\), the Kelvin temperature is \(25.0 + 273.15 = 298.15\) K.

Step 3: Calculate Initial Moles

Substitute the values into the Ideal Gas Law formula to find the initial number of moles: \( n = \frac{PV}{RT} = \frac{3.50 \times 2.00}{0.0821 \times 298.15} \). Calculate this to find \(n \approx 0.286\) moles.

Step 4: Convert the New Temperature

For the new temperature scenario, convert \(49.0^{\circ}C\) to Kelvin: \(49.0 + 273.15 = 322.15\) K.

Step 5: Calculate Moles at New Temperature

Assuming constant pressure, use the relation \( \frac{n_{1}}{T_{1}} = \frac{n_{2}}{T_{2}} \) (since \( PV = nRT \) implies \( n \propto T \) when \(P\) and \(V\) are constant). Substitute the known values: \( \frac{0.286}{298.15} = \frac{n_{2}}{322.15} \), then solve for \(n_{2}\), yielding \(n_{2} \approx 0.309\) moles.

Key Concepts

Moles Calculation

Understanding how to calculate moles is crucial when dealing with gases and chemical reactions. Moles are a way of quantifying the amount of substance using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles per mole. This concept is vital in the Ideal Gas Law, where the number of moles \(n\) plays a key role.

To calculate moles in a gas, you'll often use the Ideal Gas Law equation:

• \(PV = nRT\)
• Where \(P\) is pressure, \(V\) is volume, \(R\) is the ideal gas constant, and \(T\) is temperature.

To find the moles of oxygen gas in a tank, substitute the known values into the equation: \(n = \frac{PV}{RT}\). For instance, to find \(n\), you would plug in \(P = 3.50\) atm, \(V = 2.00\) L, \(R = 0.0821\) L atm/mol K, and \(T\) (temperature in Kelvin). Calculating this gives the number of moles, providing insight into how much of a gas there is without counting individual molecules.

Temperature Conversion

Temperature conversion is an essential step when working with gas laws because temperatures need to be in Kelvin for calculations. The Kelvin scale is preferred in scientific calculations because it begins at absolute zero, the point where molecular motion ceases. This scale directly relates temperature to energy and molecular motion.

To convert Celsius to Kelvin, simply:

• Use the formula: \(T(K) = T(°C) + 273.15\).
• For example, converting 25.0°C results in 298.15 K.

Kelvin temperatures ensure consistency in calculations, particularly in the Ideal Gas Law, which is derived from these fundamental concepts. This step might seem simple, but it's critical for accurate results, and any mistake here can lead to incorrect conclusions.

Pressure Constant Scenarios

In many gas law problems, pressure is held constant to simplify the relationship between moles and temperature. This scenario makes it easier to see how one variable affects another. When pressure and volume remain constant, the relationship between moles \(n\) and temperature \(T\) is direct, meaning as one increases, so does the other.

Under constant pressure, the Ideal Gas Law simplifies, allowing us to use

• \(\frac{n_{1}}{T_{1}} = \frac{n_{2}}{T_{2}}\).

This means the ratio of the number of moles to its temperature is constant as long as pressure and volume don't change. If you know the initial moles and temperature, you can easily find the moles at a new temperature, as demonstrated in the exercise where the temperature change from 25.0°C to 49.0°C showed an increase in moles from 0.286 to 0.309 moles. Recognizing these relationships aids in understanding how gases behave under different conditions.