In a student experiment, a constant-volume gas thermometer is calibrated in dry ice (−78.5°C ) and in boiling pentane (36.1°C). The separate pressures are 0.896 atm and 1.433 atm. Hint: Use the linear relationship P = A + BT, where A and B are constants.
What value of absolute zero does the calibration yield?
What pressure would be found at the freezing point of water?
What pressure would be found at the boiling point of water?
To find the value of absolute zero using the calibration data, we need to determine the constants A and B in the linear relationship P = A + BT.
Given the calibration points, we have two equations:
Equation 1: P1 = A + B * T1
Equation 2: P2 = A + B * T2
where T1 is the temperature of dry ice (-78.5°C) and P1 is the corresponding pressure (0.896 atm), and T2 is the boiling point of pentane (36.1°C) with P2 (1.433 atm) being the corresponding pressure.
We can rewrite both equations in terms of A and B.
Equation 1: A + B * (-78.5) = 0.896
Equation 2: A + B * 36.1 = 1.433
Now we have two linear equations with two unknowns. We can solve these equations simultaneously to find the values of A and B.
Solving Equation 1:
A - 78.5B = 0.896
Solving Equation 2:
A + 36.1B = 1.433
Multiplying Equation 1 by 36.1 and Equation 2 by 78.5 to eliminate A, we get:
36.1A - 2828.35B = 32.4256
78.5A + 2828.5B = 112.0705
Adding these two equations, we have:
114.6A = 144.4961
Dividing both sides by 114.6, we find:
A = 1.2617
Substituting A into Equation 1:
1.2617 - 78.5B = 0.896
Simplifying, we find:
-78.5B = -0.3657
Dividing both sides by -78.5, we get:
B = 0.00466
Now that we have the values of A and B, we can use the equation P = A + BT to find the values of pressures at different temperatures.
1. Given that the freezing point of water is 0°C, we can find the pressure at this temperature using Equation P = A + BT:
P = 1.2617 + 0.00466 * 0 = 1.2617 atm
2. Given that the boiling point of water is 100°C, we can find the pressure at this temperature using Equation P = A + BT:
P = 1.2617 + 0.00466 * 100 = 1.7277 atm
Therefore, the value of absolute zero obtained from the calibration is -273.15°C, the pressure at the freezing point of water is 1.2617 atm, and the pressure at the boiling point of water is 1.7277 atm.