Question

A sample containing 2.50 moles of He (1 bar, 350. K) is mixed with 1.75 mol of \(\mathrm{Ne}(1 \text { bar, } 350 . \mathrm{K})\) and \(1.50 \mathrm{mol}\) of Ar \(\left(1 \text { bar, } 350 . \text { K). Calculate } \Delta G_{\text {mixing}} \text {and } \Delta S_{\text {mixing.}}\right.\)

Short answer

The entropy change of mixing (\(\Delta S_{mix}\)) for the given mixture is approximately -120.04 J/K, and the Gibbs free energy change of mixing (\(\Delta G_{mix}\)) is approximately 42.01 kJ.

Step-by-step solution

Calculate the Mole Fractions

To find the mole fractions, we will first find the total moles of the mixture and then divide each component's moles by the total moles. Total moles = moles of He + moles of Ne + moles of Ar Total moles = 2.50 moles + 1.75 moles + 1.50 moles = 5.75 moles Mole fraction of He = moles of He / total moles = 2.50 / 5.75 Mole fraction of Ne = moles of Ne / total moles = 1.75 / 5.75 Mole fraction of Ar = moles of Ar / total moles = 1.50 / 5.75

Calculate the Entropy Change (∆S_mixing)

Now that we have the mole fractions, we can calculate the entropy change for the mixing process using the entropy change formula for ideal gases mentioned above: \(\Delta S_{mix} = -nR\sum_{i=1}^{N}x_{i}\ln{x_{i}}\) \(\Delta S_{mix}=- (5.75 \ mol)(8.314 \ J/mol \ K)[(2.50/5.75)\ln(2.50/5.75)+(1.75/5.75)\ln(1.75/5.75)+(1.50/5.75)\ln(1.50/5.75)]\) After calculating the sum and plugging in the values, we get: \(\Delta S_{mix} = -120.04 \ J/K\)

Calculate the Gibbs Free Energy Change (∆G_mixing)

Since the enthalpy change of mixing for ideal gases is zero, we can calculate the Gibbs free energy change using the formula: \(\Delta G_{mix} = \Delta H_{mix} - T\Delta S_{mix}\) \(\Delta G_{mix} = 0 - (350 \ K)(-120.04\ J/K)\) \(\Delta G_{mix}\) = 42,014 J or 42.01 kJ The entropy change of mixing is approximately -120.04 J/K and the Gibbs free energy change of mixing is approximately 42.01 kJ.

Key concepts

Entropy Change in Mixing

Entropy represents the measure of disorder or randomness in a system. When we talk about a **mixing process**, we're considering how different gases come together to form a single mixture. **Entropy Change**, noted as \(\Delta S\), shows how this disorder increases. In the exercise, helium (He), neon (Ne), and argon (Ar) gases are mixed. For ideal gases, this process is considered without any heat exchange. Hence, the **entropy change** can be calculated using the formula:\[ \Delta S_{mix} = -nR \sum_{i=1}^{N} x_i \ln x_i \]In this formula:- \(n\) is the total number of moles.- \(R\) is the ideal gas constant, 8.314 J/mol K.- \(x_i\) is the mole fraction of each gas.By calculating each gas's contribution to the entropy change and summing them up, you find the overall increase in randomness or disorder caused by mixing these gases. This is why the entropy change is negative, reflecting the process's spontaneous nature.

Understanding Ideal Gases

**Ideal Gases** are a simplified model of gases, where interactions between molecules are negligible and the volume occupied by their particles is much smaller than the free space around them. This model helps simplify calculations in chemistry. The concept of ideal gases assumes:

• All gas particles are in continuous, random motion.
• Collisions between molecules and the walls of the container are perfectly elastic, meaning total kinetic energy is conserved.
• No forces of attraction or repulsion between particles.
• Molecules are point particles with no volume.

When calculating changes in thermodynamic properties like Gibbs Free Energy or entropy for "ideal gas" mixtures, these assumptions allow us to use simplified formulas. This helps when computing the effects of mixing gases like He, Ne, and Ar, making predictions manageable and highlighting common characteristics of gas behavior without complex complications.

Calculating Mole Fraction

The **Mole Fraction** of a component in a mixture is the ratio of the number of moles of that component to the total number of moles of all components present. This concept is crucial in the mixing process of gases as seen in the exercise.To calculate the mole fraction:- First, determine the total number of moles in the mixture.For this exercise:- Total moles of gases = moles of He + moles of Ne + moles of Ar = 5.75 moles.Then, for each gas calculate:- **Mole fraction of He**: \( \frac{2.50}{5.75} \)- **Mole fraction of Ne**: \( \frac{1.75}{5.75} \)- **Mole fraction of Ar**: \( \frac{1.50}{5.75} \)These fractions help determine each gas's influence on the mixture's properties, such as its **entropy change ** and how it affects the overall Gibbs free energy.

Mixing Process in Gaseous Systems

The **Mixing Process** of gases involves combining different gases which leads to changes in thermodynamic properties like entropy and Gibbs Free Energy. The process leverages **mole fractions** to determine how each gas contributes. Mixing ideal gases is significant because:- It is spontaneous and often favorable, resulting in increased entropy or disorder. - The enthalpy change, \( \Delta H_{mix} \), is generally zero because no energy is absorbed or released.For the exercise, once mixing occurs:- Calculate the change in entropy by considering the randomness introduced by the gases intermingling (use \( \Delta S_{mix} \) formula).- Use the calculated \( \Delta S_{mix} \) to compute the Gibbs Free Energy change since \( \Delta G_{mix} = \Delta H_{mix} - T\Delta S_{mix} \). Here, \( T \) is the temperature.Understanding these changes helps predict how mixtures behave, emphasizing the role of entropy in directing spontaneous processes in chemistry.