A cylinder that has a 40.0 cm radius and is 50.0 cm deep is filled with air at 20.0°C and 1.00 atm. A 21.0 kg piston is now lowered into the cylinder, compressing the air trapped inside. Finally, a 79.0 kg man stands on the piston, further compressing the air, which remains at 20°C.
(a) How far down (Ä h) does the piston move when the man steps onto it?
(b) To what temperature should the gas be heated to raise the piston and the man back to h(initial)?
I don't know where to start from here. Do you have to a change in volume type problem? Or could you do it using the ideal gas law?
radius = .4 m so piston area = pi(.16) =.506 m^2
(a) is a change in volume problem all apparently at constant temperature = 293 K
First new volume with the piston:
P1V1/T1 = P2V2/T2
T1=T2 = 293 K
so
V2 = V1 (P1/P2)
P1 = 1 atm = 10^5 Pascals
P2 = 1 atm + (21*9.8)/.506 = 10^5+407
=100,407 Pascals
V2/V1 = 100000/100407 =.996
so
H2 = .996 (50) = 49.8 cm hardly moves
now add the man
V3 = V1(P1/P3)
P3 = 1 atm + (100*9.8)/.506 =10^5+1937
=101,937 Pascals
V3/V1 = 100000/101937 = .981
so
H3 = 50*.981 = 49.05 cm
so
when the piston was added the level went down to 49.8 and then it went down to 49.05 when the man stepped on so it went down about .8 cm due to the man.
For Part V
V2/V1 = 50/49.05 = 1.02
constant pressure so temp goes up
P1V1/T1 = P2 V2/T2
so
V2/V1 = T2/T1
T1 = 293 K
1.02 = T2/293
T2 = 298.7 K = 25.7 C
thank you so much! it's really clear now with the way you've explained it
two airplanes leave an airport at the same time. the velocity of the first airplane is 700 m/h at a heading of 52.2degrees . the velocity of the second is 600 m/h at a heading of 97 degrees . how far apart are they after 3.1 h? answer in units of m.
It was magical 🤪😃💁♀️✨
To solve this problem, we can start by using the ideal gas law to find the initial volume of air in the cylinder, before any compression occurs. The ideal gas law is given by:
PV = nRT
Where:
P = pressure (in this case, 1.00 atm)
V = volume (initial volume)
n = number of moles of gas (we can ignore this for now)
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature (in this case, 20.0°C converted to Kelvin)
We can rearrange this equation to solve for the initial volume V:
V = nRT/P
Since we are only interested in the change in volume (ΔV), we can ignore the values of n and R, since they will remain constant. So our equation becomes:
ΔV = V - V(initial) = V - nRT/P
Next, we need to determine the change in pressure (ΔP) caused by the piston and the man standing on it. This change in pressure will be directly related to the change in height (∆h) of the piston. We can use the equation for pressure:
P = F/A
Where:
P = pressure
F = force (weight of the piston + weight of the man)
A = area of the piston
Given that the weight of the piston is 21.0 kg and the weight of the man is 79.0 kg, we can calculate the total force exerted on the piston:
F = (Weight of piston) + (Weight of man) = (21.0 kg) * (9.81 m/s^2) + (79.0 kg) * (9.81 m/s^2)
Now, we have the force F and we need to find the area A of the piston. The area of a circle is given by:
A = πr^2
Where:
A = area
r = radius of the piston
Given that the radius of the cylinder is 40.0 cm (which is equivalent to 0.4 m), we can calculate the area:
A = π * (0.4 m)^2
Now that we have the force F and the area A, we can calculate the change in pressure (∆P):
ΔP = P(final) - P(initial) = (F/A) - P(initial)
Note that P(final) is the pressure after the compression caused by the piston and the man.
Finally, we can use the ideal gas law again to relate the change in volume (∆V) to the change in pressure (∆P):
ΔV = Δh * A
Where:
ΔV = change in volume
Δh = change in height of the piston
A = area of the piston
Now that we have the values for ΔV and ΔP, we can set up the equation:
ΔV = (F/A) - P(initial) * A
Solving this equation will give us the change in height (∆h) of the piston, which is the answer to part (a) of the question.
To solve part (b), we need to determine the temperature (T(final)) needed to raise the piston and the man back to the initial height. Using the ideal gas law, we rearrange the equation to solve for temperature T:
T = PV/(nR)
We know the new pressure (P(final)), the volume (V) after the compression, and we can assume the number of moles of gas (n) remains constant. Plugging in these values into the rearranged equation will give us the desired temperature (T(final)).
So, by following these steps, we can find the answers to both parts (a) and (b) of the question.