Question

A balloon contains \(160 \mathrm{L}\) of nitrogen gas at \(25^{\circ} \mathrm{C}\) and 1.0 atm. How much energy must be added to raise the temperature of the nitrogen to \(45^{\circ} \mathrm{C}\) while allowing the balloon to expand at atmospheric pressure?

Short answer

Answer: The energy required is approximately 12.7 x 10^3 J.

Step-by-step solution

In this problem, we are given the initial volume of nitrogen gas as \(160 L\), its initial temperature as \(25^\circ C\), and the initial pressure as \(1.0 atm\). We are also given that we want to raise the temperature of the nitrogen gas to \(45^\circ C\) while allowing the balloon to expand at atmospheric pressure. #Step 2: Calculating the internal energy change of the system#

The internal energy change of an ideal gas system is given by the equation \(\Delta U= n C_{v}\Delta T\), where \(\Delta U\) is the internal energy change, \(n\) is the number of moles of gas, \(C_{v}\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature. Here, \(n\) can be calculated using Ideal Gas Law: \(PV=nRT\). But before that, let's convert the given volume to SI units. Given volume \(V=160 L = 0.16 m^3\) and 1 atm in SI unit is 101325 Pa. Now using Ideal Gas Law with initial conditions, \(PV = nRT \Rightarrow n = \frac{PV}{RT}\) For nitrogen, \(R = 8.314 J/mol.K\). So we have: $n = \frac{(1.0\times 10^5 Pa)(0.16 m^3)}{(8.314 J/mol.K)(298 K)} \approx 6.52 mol$ For diatomic gases like nitrogen, \(C_{v} \approx \frac{5}{2}R\). Thus, to calculate the internal energy change, we have: $\Delta U = n C_{v}\Delta T = (6.52 mol) (\frac{5}{2}R) (45^\circ C - 25^\circ C) \approx 6.52 \times 20.8\times 10^3 J \approx 13.6 \times 10^3 J$ #Step 3: Calculating the work done by the system#

For a constant pressure process, the work done by the nitrogen gas can be expressed as: \(W = P_{external}\Delta V\) We can find \(\Delta V\) by considering the temperature change and keeping the pressure constant. \(V_2 = V_1 \frac{T_2}{T_1}\) , where \(V_1\) is the initial volume, \(V_2\) is the final volume, \(T_1\) is the initial temperature, and \(T_2\) is the final temperature. Converting temperatures to Kelvin scale, \(V_2 = (0.16 m^3)\frac{318K}{298K} \approx 0.169 m^3\) Now, calculating \(\Delta V = V_2 - V_1 \approx 0.009 m^3\) and the work done by the system: \(W = (1.0\times 10^5 Pa)(0.009 m^3) \approx 900 J\) #Step 4: Calculating the heat added to the system#

Finally, we can use the First Law of Thermodynamics (\(\Delta U = Q + W\)) to calculate the heat added to the system, \(Q\): \(Q = \Delta U - W \approx (13.6 \times 10^3 J) - (900 J) \approx 12.7 \times 10^3 J\) Thus, the energy that must be added to raise the temperature of the nitrogen gas to \(45^\circ C\) while allowing the balloon to expand at atmospheric pressure is approximately \(12.7 \times 10^3 J\).