Problem 4 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Estimate the change in entropy when 1.00 mol of argon is heated from \(300 \mathrm{K}\) to \(1200 \mathrm{K}\). What assumptions have you made and how could you make your estimate more accurate? (Section 14.2 ) (The heat capacity, \(C_{p^{\prime}}\), of argon gas is \(20.8 \mathrm{JK}^{-1}\) mol \(^{-1}\).)

Step-by-Step Solution

Answer

Entropy change is approximately 28.80 J/K. Assumed constant heat capacity for argon.

Step 1: Understand the concept of entropy change

Entropy is a measure of the amount of energy in a system that is not available for doing work. When a substance is heated, its entropy increases because the molecules move more vigorously. To find the change in entropy when a mol of argon is heated from one temperature to another, we can use the formula for entropy change due to temperature change: \[ \Delta S = n C_p \ln \left(\frac{T_2}{T_1}\right), \]where \(\Delta S\) is the change in entropy, \(n\) is the number of moles, \(C_p\) is the heat capacity at constant pressure, \(T_1\) is the initial temperature, and \(T_2\) is the final temperature.

Step 2: Apply the formula for entropy change

We know that \(n = 1.00\, \text{mol}\), \(C_p = 20.8 \, \text{J/K.mol}\), \(T_1 = 300 \, \text{K}\), and \(T_2 = 1200 \, \text{K}\). Substitute these values into the formula:\[ \Delta S = 1.00 \, \text{mol} \times 20.8 \, \text{J/K.mol} \times \ln \left(\frac{1200}{300}\right). \]

Step 3: Calculate the natural logarithm

Calculate the natural logarithm part of the equation: \[ \ln \left(\frac{1200}{300}\right) = \ln (4). \]

Step 4: Compute \(\ln(4)\)

The value of \(\ln(4)\) can be calculated using a calculator or logarithmic tables:\[ \ln(4) \approx 1.386. \]

Step 5: Substitute and solve for \(\Delta S\)

Now substitute \(\ln(4) = 1.386\) back into the equation and solve for \(\Delta S\): \[ \Delta S = 1.00 \times 20.8 \times 1.386 \approx 28.80 \, \text{J/K}. \]

Step 6: Consider assumptions and accuracy

One assumption made in this calculation is that the heat capacity \(C_p\) of argon is constant over the temperature range from 300 K to 1200 K. To make the estimate more accurate, one could use temperature-dependent heat capacity values or integrate the heat capacity function over the temperature range if such data is available.

Key Concepts

Entropy

Entropy is an essential concept in thermodynamics that represents the degree of disorder or randomness in a system. When a system undergoes a process, such as heating, its entropy typically changes. For gases, as they are heated, their molecules begin to move more vigorously, leading to an increase in entropy. This is because the energy that was previously used for work becomes more spread out and less available.
To quantify this change in entropy during a temperature change, we use the formula:

• \(\Delta S = n C_p \ln \left(\frac{T_2}{T_1}\right)\)

Here \(\Delta S\) indicates the change in entropy, \(n\) is the number of moles, \(C_p\) is the heat capacity at constant pressure, and \(T_1\) and \(T_2\) are the initial and final temperatures, respectively. Understanding how entropy functions in a system can help determine the feasibility and spontaneity of a thermodynamic process.

Heat Capacity

Heat capacity, particularly the specific heat capacity at constant pressure, \(C_p\), is a measure of the amount of heat required to change a substance's temperature by one degree Kelvin per mole. In other words, it tells us how much heat energy is needed to raise the temperature of a mole of gas.
For argon gas, the given heat capacity is \(C_p = 20.8 \text{ J/K.mol}\). This means that for every mole of argon, 20.8 joules are required to increase the temperature by 1 Kelvin when pressure is kept constant.
In calculations of entropy change during heating, using a constant \(C_p\) simplifies the computation. However, for more precision, one might consider that \(C_p\) can vary with temperature, which could significantly affect the calculated entropy change.

Temperature Change

Temperature change is a core factor in processes that involve heat exchange and the accompanying changes in entropy. In this case, we're interested in how heating affects entropy when argon gas is heated from an initial temperature \(T_1 = 300 \text{ K}\) to a final temperature \(T_2 = 1200 \text{ K}\).
The extent of temperature change directly influences the increase in thermal energy within the system, thus affecting the entropy. As temperature rises, molecules become more energetic and disordered, and hence the entropy increases.
This is why in our formula for \(\Delta S\), the temperature change is represented as the ratio \(\frac{T_2}{T_1}\). The natural logarithm of this ratio forms part of the calculation for entropy change, emphasizing how even a modest temperature rise can significantly impact the entropy.

Moles of Gas

The number of moles of a substance, \(n\), is crucial in determining the total change in entropy during a thermal process. It essentially scales the impact of heat transfer by specifying how much gas is involved in the process.
In our scenario with argon gas, \(n = 1.00\text{ mol}\), which means the calculations for entropy change are based on a single mole of argon being heated.
Choosing the right number of moles is important, as it directly influences the outcome of the entropy change calculation. For multiple moles, the entropy change would be proportionally larger, following the direct relationship in the formula \(\Delta S = n C_p \ln \left(\frac{T_2}{T_1}\right)\).
Thus, understanding the concept of moles helps in quantifying and predicting the thermodynamic changes occurring in a chemical system.