What is the approximate energy required to raise the temperature of 5 L of air by 230 °C? The volume is held constant. (Assume air almost entirely consists of nitrogen, oxygen, and argon, and that it is initially at 273 K.
Cv for nitrogen=20.7,oxygen=21 and argon=12.5
percentage by volume for nitrogen is 78.08, oxygen is 20.95 and argon is 0.93
To calculate the energy required to raise the temperature of a given volume of air, we can use the equation:
Q = m * Cv * ΔT
Where:
Q = Energy required (in Joules)
m = Mass of the air (in kg)
Cv = Specific heat capacity at constant volume for the mixture of gases (in J/kg·K)
ΔT = Change in temperature (in °C or K)
First, let's calculate the mass of the air in the given volume:
Mass = Volume * Density
Since the density of air is approximately 1.225 kg/m³, the mass of 5 L of air can be calculated as:
Mass = 5 L * (1 m³ / 1000 L) * 1.225 kg/m³
Now that we have the mass, let's calculate the energy required by each gas component (N₂, O₂, and Ar):
For nitrogen (N₂):
Q_N₂ = (Mass_N₂) * (Cv_N₂) * ΔT
For oxygen (O₂):
Q_O₂ = (Mass_O₂) * (Cv_O₂) * ΔT
For argon (Ar):
Q_Ar = (Mass_Ar) * (Cv_Ar) * ΔT
Finally, add up the energy required for each gas component to get the total energy required:
Q_total = Q_N₂ + Q_O₂ + Q_Ar
Let's calculate the values step by step:
Step 1: Calculate the mass of air
Mass_air = 5 L * (1 m³ / 1000 L) * 1.225 kg/m³
Step 2: Calculate the energy required for each gas component
Q_N₂ = (Mass_air) * (Cv_N₂) * ΔT
Q_O₂ = (Mass_air) * (Cv_O₂) * ΔT
Q_Ar = (Mass_air) * (Cv_Ar) * ΔT
Step 3: Calculate the total energy required
Q_total = Q_N₂ + Q_O₂ + Q_Ar
To calculate the approximate energy required to raise the temperature of a given volume of air, we need to use the specific heat capacity (Cv) for each constituent gas present, as well as the initial and final temperatures.
First, we need to calculate the mass of each gas in the given volume of air. Since the volume is held constant, the mass of each gas remains the same throughout the temperature change. We multiply the volume (5 L) by the percentage by volume for each gas to find the volume of each gas in liters.
For nitrogen:
Volume of nitrogen = 5 L * 0.7808 = 3.904 L
For oxygen:
Volume of oxygen = 5 L * 0.2095 = 1.048 L
For argon:
Volume of argon = 5 L * 0.0093 = 0.0465 L
Next, we need to convert the volumes of the gases to moles using the ideal gas law equation. At 273 K and 1 atm pressure, one mole of an ideal gas occupies 22.4 L.
For nitrogen:
Moles of nitrogen = 3.904 L / 22.4 L/mol = 0.174 moles
For oxygen:
Moles of oxygen = 1.048 L / 22.4 L/mol = 0.047 moles
For argon:
Moles of argon = 0.0465 L / 22.4 L/mol = 0.00208 moles
Now that we have the moles of each gas, we can calculate the energy required to raise the temperature of each gas using the specific heat capacity (Cv) for each gas.
For nitrogen:
Energy for nitrogen = Moles of nitrogen * Cv for nitrogen * ΔT = 0.174 moles * 20.7 J/(mol·K) * 230 °C
For oxygen:
Energy for oxygen = Moles of oxygen * Cv for oxygen * ΔT = 0.047 moles * 21 J/(mol·K) * 230 °C
For argon:
Energy for argon = Moles of argon * Cv for argon * ΔT = 0.00208 moles * 12.5 J/(mol·K) * 230 °C
Finally, we sum up the energy required for each gas to obtain the total energy required to raise the temperature of the air:
Total energy = Energy for nitrogen + Energy for oxygen + Energy for argon
By substituting the values into the equations and performing the calculations, you will obtain the approximate energy required.