I once was given a metal whiskey flask for being in my cousin’s wedding. When it got dented, I put in 1 gram of H2O (initially 21°C, just like the air filling the rest of the flask) and heated it up to boil the water. It popped the dent out.
Assume the flask had a rectangular cross section, with a height of 15 cm, a width of 10 cm, and a thickness of 2 cm. Assume that in order to pop the dent out, an outward pressure difference of 1.0E6 Pa was required (that is, the outward pressure on the flask minus the inward pressure on the flask was 1.0E6 Pa). How much heat had to be added to the contents of the flask?
At the instant the dent popped out, was the gas inside the flask doing positive work, or was positive work being done to the gas?
To find out how much heat had to be added to the contents of the flask, we can use the ideal gas law and the concept of work done.
First, let's calculate the initial volume of the flask:
The flask has a rectangular cross-section, so the initial volume can be found by multiplying the width, height, and thickness of the flask:
Volume = width x height x thickness = 10 cm x 15 cm x 2 cm = 300 cm^3
Since the flask is initially filled with air and has the same temperature as the water, we can assume the air is ideal gas (although this assumption may not be entirely accurate).
Now, let's determine the change in pressure required to pop the dent out:
The pressure difference needed to pop the dent is given as 1.0E6 Pa (outward pressure minus inward pressure). This is the pressure exerted by the gas inside the flask.
Next, we need to calculate the change in volume that occurs due to the dent popping out:
The dent popping out increases the volume of the flask, resulting in an increase in the volume of the gas inside. We can calculate this change in volume by using the initial volume of the flask and the change in pressure:
Change in volume = (pressure difference * initial volume) / initial pressure
Assuming the initial pressure of the gas inside the flask is approximately equal to the atmospheric pressure (which is around 1.0E5 Pa), we can substitute the values into the equation:
Change in volume = (1.0E6 Pa * 300 cm^3) / (1.0E5 Pa) = 3000 cm^3
Now, we have the change in volume of the gas inside the flask. To find the amount of heat added to the contents of the flask, we can use the formula for the first law of thermodynamics:
Q = ΔU + W
Where Q is the heat added, ΔU is the change in internal energy, and W is the work done. Since the flask is open to the atmosphere and considered an isolated system, the change in internal energy (ΔU) is zero. Therefore, the equation simplifies to:
Q = W
To determine whether positive work is being done by the gas or positive work is being done to the gas, we need to consider the sign convention for work.
In this scenario, when the dent pops out, the volume of the gas increases. As a result, the gas is doing work on its surroundings (positive work is being done by the gas).
Therefore, the amount of heat (Q) added to the contents of the flask is equal to the work (W) done by the gas, which is the change in pressure multiplied by the change in volume:
Q = W = (pressure difference) * (change in volume)
Q = 1.0E6 Pa * 3000 cm^3
Now you can calculate Q by converting cm^3 to m^3 and performing the multiplication.
Note: For a more precise calculation and to account for real gas behavior, one would need to consider the specific properties of the gas inside the flask and use a proper equation of state, such as the Van der Waals equation.