Question

Consider a balloon filled with helium at the following conditions. $$ \begin{array}{l} 313 \text { g He } \\ 1.00 \text { atm } \\ \text { 1910. L } \\ \text { Molar Heat Capacity }=20.8 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{mol} \end{array} $$ The temperature of this balloon is decreased by \(41.6^{\circ} \mathrm{C}\) as the volume decreases to \(1643 \mathrm{~L}\), with the pressure remaining constant. Determine \(q, w\), and \(\Delta E\) (in \(\mathrm{kJ}\) ) for the compression of the balloon.

Short answer

For the compression of the helium-filled balloon, the calculated values are: heat, \(q = -68.1 \: kJ\); work, \(w = 267 \: kJ\); and change in internal energy, \(\Delta E = 198.9 \: kJ\).

Step-by-step solution

Calculate the moles of helium

Given the mass of helium (313g) and the molar mass of helium (4.00 g/mol), we can calculate the moles (n) of helium in the balloon: \[ n = \frac{mass_{He}}{MolarMass_{He}} \] \[ n = \frac{313}{4.00} = 78.25 mol \]

Find the initial and final temperature

We will use the ideal gas law for both initial and final states to find the initial and final temperatures: \( P_1V_1 = nRT_1 \) and \(P_2V_2 = nRT_2 \) Given: \( P_1 = P_2; V_1 = 1910 L; V_2 = 1643 L; n = 78.25 mol; R = 0.0821 \frac{L.atm}{K.mol} \) \[ T_1 = \frac{P_1V_1}{nR} \] \[ T_1 = \frac{(1.00)(1910)}{(78.25)(0.0821)} = 298.6 K \] The temperature is decreased by \(41.6^{\circ}C\), so: \[ \Delta T = -41.6 K \] \[ T_2 = T_1 + \Delta T = 298.6 K - 41.6 K = 257.0 K \]

Calculate heat (q)

We will use the formula: \[ q = n\times C_{p} \times \Delta T \] Where, \( q \) is the heat, \( n \) is the number of moles, \( C_{p} \) is the molar heat capacity, and \( \Delta T \) is the change in temperature. \[ q = (78.25) \times (20.8) \times (-41.6) = -68,130 J \] Convert this to kJ: \[ q = -68.1 kJ \]

Calculate work (w)

The formula for work (w) is given by: \[ w = -P \times \Delta V \] Where, \(P\) is the pressure, and \(\Delta V\) is the change in volume. \[ \Delta V = V_2 - V_1 = 1643 - 1910 = -267 L \] \[ w = -(1.00) \times (-267) = 267 kJ \]

Calculate change in internal energy (ΔE)

We can calculate the change in internal energy (ΔE) using the formula: \[ \Delta E = q + w \] \[ \Delta E = (-68.1) + (267) = 198.9 kJ \] The values of heat (q), work (w), and change in internal energy (ΔE) for the compression of the balloon are: \[ q = -68.1 \: kJ \] \[ w = 267 \: kJ \] \[ \Delta E = 198.9 \: kJ \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial formula in thermodynamics that relates the pressure, volume, temperature, and amount of gas in a system. It is expressed as \(PV = nRT\), where:

• \(P\) stands for pressure.
• \(V\) is volume.
• \(n\) represents the number of moles.
• \(R\) is the gas constant, with a value of 0.0821 \(\frac{L.atm}{K.mol}\) for these units.
• \(T\) is the temperature, measured in Kelvin.

The Ideal Gas Law helps us understand how gases will behave under different conditions by mathematically relating these parameters. For example, if we have a set amount of a gas and increase its temperature, we can expect the volume or pressure to change accordingly, depending on the situation. In the exercise, this law was used twice to find the initial and final temperatures by rearranging the formula to \(T = \frac{PV}{nR}\). It’s a versatile equation essential for understanding gas behaviors in many scientific problems.

Molar Heat Capacity

Molar Heat Capacity \(C_p\), in this exercise, is used to calculate the amount of heat involved during a temperature change of a gas. It refers to the amount of energy required to raise the temperature of one mole of a substance by one degree Celsius. The provided molar heat capacity was 20.8 \(J/^{\circ}C \. mol\).

• It’s important to distinguish between the heat capacities at constant volume \(C_v\) and constant pressure \(C_p\).
• \(C_p\) is typically used when working with gases that can expand or contract, as seen in this exercise.

In the step-by-step solution, the formula \(q = n \times C_p \times \Delta T\) was used, where \(q\) is the heat energy transferred. This formula tells us how temperature change will affect the energy of a gas system, highlighting the role of molar heat capacity in thermodynamics.

Internal Energy Change

The internal energy change \(\Delta E\) of a system indicates how the energy within a system changes as work is performed or heat is transferred. It is linked to the First Law of Thermodynamics, which states that energy cannot be created or destroyed, only transferred or transformed.

• The formula \(\Delta E = q + w\) is used, where \(q\) is the heat added to the system and \(w\) is the work done by or on the system.
• It helps to understand how energy balances in scenarios involving heat transfer and work, such as expanding or compressing a gas.

In the given exercise, the values computed for heat \(q\) and work \(w\) were plugged into this equation, showing how energy shifts within the balloon during compression. This calculation illustrates the conservation of energy within thermodynamic processes.