A scuba tank contains 1.5 kg. of air. The air in the tank is initially at 15 degree celcius. The tank is left near an engine exhaust line, and the tank's pressure doubles. Determine the final temperature, change of internal energy, change of enthalpy, and heat.
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To solve this problem, we need to apply the ideal gas law and consider the relationship between temperature, pressure, and volume. The ideal gas law can be expressed as:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
Given:
Initial mass of air (m) = 1.5 kg
Initial temperature (T1) = 15°C = 15 + 273.15 K (Convert to Kelvin)
Final pressure (P2) = double the initial pressure = 2P1
Step 1: Convert the initial temperature to Kelvin
T1 = 15 + 273.15 = 288.15 K
Step 2: Calculate the number of moles of air (n1) using the mass and molar mass of air
Molar mass of air = 28.97 g/mol
n1 = (m/M)
n1 = (1.5 kg / 0.02897 kg/mol) = 51.84 mol (rounded to two decimal places)
Step 3: Use the ideal gas law to calculate the initial volume (V1)
R = 8.314 J/(mol·K) (ideal gas constant)
V1 = (n1RT1) / P1
Step 4: Calculate the final volume (V2) using Boyle's Law, as pressure is doubled while the amount of air stays constant
V1 * P1 = V2 * P2
V2 = (V1 * P1) / P2
Step 5: Calculate the final temperature (T2) using the ideal gas law
T2 = (n1R * T1) / V2
Step 6: Calculate the change in internal energy (ΔU) using the equation:
ΔU = n1Cv(T2 - T1)
Cv is the heat capacity at constant volume. For diatomic gases, Cv = 5/2 R.
Step 7: Calculate the change in enthalpy (ΔH) using the equation:
ΔH = ΔU + P2 * (V2 - V1)
Step 8: Calculate the heat exchanged (Q) using the first law of thermodynamics:
Q = ΔU - W (change in internal energy minus work done)
Now we can calculate the final temperature, change in internal energy, change in enthalpy, and heat:
Step 1: Convert the initial temperature to Kelvin
T1 = 15 + 273.15 = 288.15 K
Step 2: Calculate the number of moles of air
n1 = (m/M)
n1 = (1.5 kg / 0.02897 kg/mol) = 51.84 mol (rounded to two decimal places)
Step 3: Calculate the initial volume
V1 = (n1RT1) / P1
V1 = (51.84 mol * 8.314 J/(mol·K) * 288.15 K) / P1
Step 4: Calculate the final volume
V2 = (V1 * P1) / P2
Step 5: Calculate the final temperature
T2 = (n1R * T1) / V2
Step 6: Calculate the change in internal energy
ΔU = n1Cv(T2 - T1)
Cv = 5/2 R
Cv = (5/2) * 8.314 J/(mol·K)
ΔU = 51.84 mol * [(5/2) * 8.314 J/(mol·K)] * (T2 - T1)
Step 7: Calculate the change in enthalpy
ΔH = ΔU + P2 * (V2 - V1)
Step 8: Calculate the heat exchanged
Q = ΔU - W
Note: To calculate the final volume, pressure (P1) and pressure (P2) must be in the same units. The ideal gas law equation will use consistent units.
To solve this problem, we need to apply the ideal gas law and the first law of thermodynamics. Here's how we can calculate the final temperature, change of internal energy, change of enthalpy, and heat:
Step 1: Convert the mass of air in the tank from kilograms to moles.
The molar mass of air (approximately) is 28.97 g/mol. Therefore, the number of moles of air in the tank can be calculated as follows:
moles = mass / molar mass
moles = 1500 g / 28.97 g/mol
moles = 51.8 mol (approximately)
Step 2: Apply the ideal gas law to find the initial pressure.
The ideal gas law is expressed as:
PV = nRT,
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
Since the volume of the tank is not given, we'll assume it remains constant. Hence, we can write the initial pressure as:
P1V = nRT1
Step 3: Use the initial temperature of 15 degrees Celsius to convert it to Kelvin.
T1 = 15 + 273.15 = 288.15 K
Step 4: Double the initial pressure to find the final pressure.
P2 = 2 * P1
Step 5: Apply the ideal gas law to find the final temperature.
P2V = nRT2
Since the volume and number of moles haven't changed, and we know the final pressure and the gas constant R, we can rearrange the equation to solve for T2:
T2 = (P2 * V) / (n * R)
Step 6: Calculate the change of internal energy (ΔU).
The change in internal energy can be found using the first law of thermodynamics:
ΔU = Q - W
However, in this case, there is no work done on or by the system, so W = 0.
Therefore, ΔU = Q.
Step 7: Calculate the change of enthalpy (ΔH).
The change in enthalpy can be calculated using the equation:
ΔH = ΔU + PΔV
Since the volume remains constant, ΔV = 0.
Therefore, ΔH = ΔU
Step 8: Calculate the heat (Q) using the equation:
Q = nCΔT
Where C is the molar heat capacity of air at constant pressure.
Now, let's do the calculations:
- Step 1: Mass of air in the tank = 1.5 kg
- Step 2: Calculate the number of moles (moles = mass / molar mass): 51.8 mol
- Step 3: Convert the initial temperature of 15°C to Kelvin: T1 = 288.15 K
- Step 4: Double the initial pressure: P2 = 2 * P1
- Step 5: Calculate the final temperature: T2 = (P2 * V) / (n * R)
- Step 6: Calculate the change in internal energy (ΔU): ΔU = Q
- Step 7: Calculate the change in enthalpy (ΔH): ΔH = ΔU
- Step 8: Calculate the heat (Q): Q = nCΔT
Note: To obtain the values for R (ideal gas constant) and C (molar heat capacity), you need to refer to the specific values for air.