Question

Treating air as an ideal gas of diatomic molecules, calculate how much heat is required to raise the temperature of the air in an \(8.00 \mathrm{~m}\) by \(10.0 \mathrm{~m}\) by \(3.00 \mathrm{~m}\) room from \(20.0^{\circ} \mathrm{C}\) to \(22.0^{\circ} \mathrm{C}\) at \(101 \mathrm{kPa}\). Neglect the change in the number of moles of air in the room.

Short answer

The room dimensions are 8.00 m in length, 10.0 m in width, and 3.00 m in height. Answer: The heat required to raise the temperature of the air in the room from \(20.0^{\circ}\mathrm{C}\) to \(22.0^{\circ}\mathrm{C}\) at 101 kPa is approximately 41011.4 J.

Step-by-step solution

Calculate the volume of the room

Calculate the volume V of the room (in \(m^3\)) by multiplying its length, width, and height. The volume V can be calculated as: \(V = length \times width \times height\) Here, length = 8.00 \(m\), width = 10.0 \(m\), and height = 3.00 \(m\). So, V = 8.00 \(\times\) 10.0 \(\times\) 3.00 = 240 \(m^3\)

Calculate the number of moles of air in the room

Use the ideal gas law to calculate the number of moles of air (n) in the room, given the pressure (P), volume (V) and temperature (T). The ideal gas law is written as: \(PV = nRT\), where R is the ideal gas constant (8.314 \(J/mol K\)). First, convert the initial temperature \(T_1 = 20.0^{\circ}\mathrm{C}\) to Kelvin by adding 273.15 to it. We get \(T_1 = 293.15\) K. The pressure (P) of the room is given as 101 kPa, which we need to convert it into Pa. So, \(P = 101 \times 10^3\) Pa. Now, rewrite the ideal gas law equation to find the number of moles: \( n = \frac{PV}{RT} \) Plug in the known values: \(n = \frac{101 \times 10^3 \times 240}{8.314 \times 293.15}\) Solve for n: \(n \approx 987.044\) moles.

Calculate the heat capacity of the air

For a diatomic ideal gas, the molar heat capacity at constant volume (Cv) is given by: \(Cv = \frac{5}{2}R\), where R is the ideal gas constant. Calculate the molar heat capacity Cv: \(Cv = \frac{5}{2} \times 8.314 = 20.785 \, J/mol K\) Now we will find the total heat capacity for the room by multiplying Cv and the number of moles n. Q = n * Cv Q = 987.044 * 20.785 »

Calculate the heat required

To calculate the heat required to increase the temperature of the air (ΔQ), we will use the formula: \(\Delta Q = Q \Delta T\), where ΔT is the change in temperature. First, calculate the change in temperature ΔT: Final temperature, \(T_2 = 22.0^{\circ}\mathrm{C} => T_2 = 295.15\) K (after converting to Kelvin) ΔT = \(T_2 - T_1\) ΔT = 295.15 - 293.15 ΔT = 2 K Now, calculate the heat required to raise the temperature by 2 K: \(\Delta Q = Q \Delta T = 987.044 \times 20.785 \times 2\) Solve for ΔQ: \(\Delta Q \approx 41011.4\, J\) So, the heat required to raise the temperature of the air in the room from \(20.0^{\circ}\mathrm{C}\) to \(22.0^{\circ}\mathrm{C}\) at 101 kPa is approximately 41011.4 J (Joules).

Key concepts

Ideal Gas Law

Understanding the ideal gas law is crucial when dealing with problems concerning gas behavior under certain conditions of temperature, volume, and pressure. This law is represented by the equation \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles of the gas, R is the ideal gas constant (equal to 8.314 \(J/mol\text{ }K\)), and T is the temperature in Kelvin.
When treating air as an ideal gas, the initial step is often to calculate the number of moles present in a given volume at a specific pressure and temperature, which was performed using the ideal gas law in the exercise. It's important to realize that this law assumes ideal conditions, meaning no intermolecular forces between particles and that the particles occupy no volume themselves. However, for many applications, including the problem at hand, the ideal gas law provides a sufficiently accurate model for the behavior of gases like air.

Molar Heat Capacity

The molar heat capacity is an essential concept that reflects the amount of heat needed to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). For gases, the molar heat capacity can differ depending on whether the heat is added at constant volume (\(C_v\)) or at constant pressure (\(C_p\)).
In our exercise, we focus on \(C_v\), the molar heat capacity at constant volume, especially because the heat was added to air in an enclosed room. The formula for \(C_v\) for diatomic gases, such as air, is \(\frac{5}{2}R\), illustrating a direct relationship with the ideal gas constant, R. This value accounts for the translational and rotational kinetic energy modes that diatomic molecules can possess. The calculation of the total heat capacity involves multiplying \(C_v\) by the number of moles to determine the overall heat capacity of the air in the room.

Temperature Conversion

Temperature conversion is a vital step when working with the ideal gas law, as temperature needs to be in Kelvin. Celsius and Kelvin are on the same scale with a difference of 273.15, meaning to convert from Celsius to Kelvin, you add 273.15 to the Celsius temperature.
In the context of the exercise, the conversion was applied to the initial room temperature of \(20.0^{\text{\degree}C}\) to get \(293.15 K\) and the final room temperature of \(22.0^{\text{\degree}C}\) to reach \(295.15 K\). Getting the correct temperature in Kelvin is pivotal in accurately applying the ideal gas law, as using degrees Celsius would not provide meaningful results with the absolute scale required by thermodynamic equations.

Diatomic Molecules

Diatomic molecules, which are composed of two atoms, exhibit distinctive characteristics regarding heat capacity. When treating air as an ideal gas, we consider it to be made of diatomic molecules, predominantly oxygen (O2) and nitrogen (N2).
Their kinetic energy can be found in multiple forms: translational, rotational, and, to a lesser degree, vibrational. In the context of this problem, these kinetic energy modes are considered in the calculation of the molar heat capacity at constant volume. The molecular structure of diatomic molecules, particularly their ability to rotate and vibrate, plays a role in determining the heat capacity. Understanding the behavior of diatomic molecules, which make up most of the Earth's atmosphere, is essential for predicting how gases will respond to changes in energy in the form of heat.