Imagine a 600 g of water at 70oC in a piston-cylinder device. Initially, the 9-kg, 13-cm diameter piston rests on two stops such that the water occupies a volume of 0.2 m3. Heat is transferred to the system until the water is completely vaporized. Take the atmospheric pressure to be 1 atm.
To what temperature should the water be heated to lift the piston?
To find the temperature to which the water should be heated in order to lift the piston, we need to consider the equilibrium between the pressure exerted by the water and the weight of the piston.
Given:
Mass of water (m) = 600 g = 0.6 kg
Initial volume of water (V) = 0.2 m³
Pressure at atmospheric pressure (P₀) = 1 atm
Mass of piston (M) = 9 kg
Diameter of piston (d) = 13 cm = 0.13 m
We can start by calculating the initial pressure exerted by the water using the ideal gas law. Since the water is in a piston-cylinder device, we can assume it behaves like an ideal gas. The ideal gas law equation is:
PV = nRT
Where:
P is the pressure,
V is the volume,
n is the number of moles of the gas (which can be calculated using the given mass and molar mass of water),
R is the ideal gas constant,
T is the temperature.
We can rearrange the equation to solve for the initial pressure P₀:
P₀ = (n₀RT₀) / V₀
Next, we need to find the final volume of the water when it completely vaporizes. At that point, the piston will be lifted, and the volume will have increased to accommodate the water vapor.
To calculate the final volume, we use the relationship between the diameter and the volume of a cylinder:
V = πr²h
Where:
V is the volume,
r is the radius,
h is the height.
Since the diameter is given, we can calculate the initial radius (r₀) using the formula:
r₀ = d / 2
To find the final radius (rₑ), we use the fact that the volume of the water is conserved. Hence, we equate the initial volume to the final volume:
V₀ = πrₑ²h
Finally, we can calculate the final temperature (Tₑ) of the water using the ideal gas law. The number of moles of water in the final state will be the same as initially, but the volume will have changed:
Pₑ = (n₀RTₑ) / Vₑ
We can rearrange this equation to solve for the final temperature Tₑ:
Tₑ = (Pₑ Vₑ) / (n₀R)
Now, let's go through the calculations to find the temperature to which the water should be heated to lift the piston.