Question

What is the value of \(\triangle \mathrm{E}\), when \(64 \mathrm{~g}\) oxygen is heated from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) at constant volume? \(\left(\mathrm{C}_{\mathrm{v}}\right.\) on an average is \(5 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) ): (a) \(1500 \mathrm{~J}\) (b) \(1800 \mathrm{~J}\) (c) \(2000 \mathrm{~J}\) (d) \(2200 \mathrm{~J}\)

Short answer

The value of \(\Delta E\) is 1000 J.

Step-by-step solution

Calculate Moles of Oxygen

To find the number of moles of oxygen, we need to use the molar mass of oxygen. Since oxygen (O_2) has a molar mass of 32 g/mol (16 g/mol for each oxygen atom), 64 g of oxygen is equivalent to \(\frac{64}{32} = 2\) moles.

Calculate Change in Internal Energy

The change in internal energy \(\Delta E\) can be calculated using the formula: \( \Delta E = n C_v \Delta T \), where \(n\) is the number of moles, \(C_v\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature. So, using \(n = 2\) moles, \(C_v = 5 \text{ JK}^{-1}\text{mol}^{-1}\), and \(\Delta T = 100^{\circ}C - 0^{\circ}C = 100\,\text{K}\), we find \( \Delta E = 2 \times 5 \times 100 = 1000\, \text{J}\).

Assess Any Mistakes and Correct

Recall that \(C_v = 5 \text{ JK}^{-1}\text{mol}^{-1}\) was given as an average, so we must ensure it's multiplied correctly by \(\Delta T\). Verify calculations: \(\Delta E = 2 \text{ moles} \times 5 \text{ JK}^{-1}\text{ mol}^{-1} \times 100 \text{ K} = 1000 \text{ J}\) calculated internally once, reassess and correct if necessary.

Key concepts

Change in Internal Energy

In thermodynamics, the change in internal energy (\(\Delta E\)) of a system is an important concept. It corresponds to the energy change within a system due to heat transfer, work done, or both. When a gas is heated, like in the case of our oxygen at constant volume, the internal energy of the gas changes. This change can be calculated using a specific formula:

• \(\Delta E = n C_v \Delta T\)

Where:

• \(n\) is the number of moles,
• \(C_v\) is the molar heat capacity at constant volume, and
• \(\Delta T\) is the change in temperature.

In our example, oxygen's internal energy changes because the gas is uniformly heated from \(0^{\circ}C\) to \(100^{\circ}C\). This change is independent of the path taken and solely depends on the initial and final states.

Molar Heat Capacity

The molar heat capacity at constant volume, represented by \(C_v\), is the amount of heat energy required to raise the temperature of one mole of a substance by one Kelvin, with the volume held constant. Here, for the oxygen gas, an average value of \(C_v = 5 \, \text{JK}^{-1}\, \text{mol}^{-1}\) is used.This concept helps us understand how different substances absorb energy. A higher molar heat capacity means that more energy is needed to achieve the same temperature change compared to a substance with a lower \(C_v\). Remember, the heat capacity can vary depending on conditions and is often given as an average like in our exercise. Using constant values simplifies calculations by assuming no specific heat interactions change over small temperature ranges.

Moles Calculation

Calculating the number of moles is a fundamental aspect of solving thermodynamic problems involving gases. Moles provide a bridge between the mass of a substance and its molecular composition. Using the relationship:

• \(n = \frac{\text{mass}}{\text{molar mass}}\)

we determined the moles of oxygen gas. Given that the molar mass of oxygen (\(O_2\)) is \(32\, \text{g/mol}\) (since each atom has a molar mass of \(16\, \text{g/mol}\)), 64 grams of oxygen translates to:\[ n = \frac{64\, \text{g}}{32\, \text{g/mol}} = 2\, \text{moles}\]This conversion is crucial as all further calculations for internal energy rely on the correct number of moles. Always remember to use accurate molar masses for precision in your problem-solving.

Temperature Change

Temperature change (\(\Delta T\)) is simply the difference between the final and initial temperatures of a system. In thermodynamics, it is one of the most straightforward measures to track energy changes as it directly relates to the state of the system. For our oxygen problem, the gas heats from \(0^{\circ}C\) to \(100^{\circ}C\), resulting in a temperature change:\[\Delta T = 100^{\circ}C - 0^{\circ}C = 100\, \text{K}\]It's crucial to understand that a change in Celsius is numerically equal to a change in Kelvin, hence why the conversion isn't necessary for the difference. Such temperature changes are pivotal in controlling the calculations involving energy, making it possible to eventually find the change in internal energy using the related formula. Understanding temperature change is essential in relating physical changes to thermodynamic quantities.