The gases in a hair spray can are at a temperature of 27 C and a pressure of 30 lbs/in^2. If the gases in the can reach a pressure of 90 lbs'in^2, the can will explode. To what temperature must the gases be raised to in order for the can to explode? Assume constant volume. (630 degrees C) The answer is 900K. Show your work to solve the problem.
P1/T1 = P2/T2 or
P1*T2 = P2*T1
T1 = 27 C = 273 + 27 = 300 kelvin
T2 = ? K
P1 = 30 lbs/in^2
P2 = 90 lbs/in^2
Substitute and solve for T2 in kelvin.
Post your work if you get stuck.
loe
To solve this problem, we can use Charles' Law, which states that the volume of a gas is directly proportional to its temperature at constant pressure.
Let's denote V1, T1, and P1 as the initial volume, temperature, and pressure of the gas, respectively. Similarly, denote V2, T2, and P2 as the final volume, temperature, and pressure of the gas when the can explodes.
According to the problem, we have:
T1 = 27°C (or 300K)
P1 = 30 lbs/in^2
P2 = 90 lbs/in^2 (given)
Since the volume remains constant, we can write:
(V1 / T1) = (V2 / T2)
The initial and final volumes (V1 and V2) are the same, so we can cancel them out:
(T1 / T2) = 1
Now, let's rewrite the equation with the given values:
(300K / T2) = 1
To find T2, we can rearrange the equation:
T2 = 300K
Therefore, the gases in the can must be raised to a temperature of 900K (or 630°C) in order for the can to explode.
To solve this problem, we can use the combined gas law, which relates the initial and final conditions of temperature and pressure of a gas at constant volume. The combined gas law is expressed as:
(P1 * T1) / (P2 * T2) = k
Where P1 and P2 are the initial and final pressures, T1 and T2 are the initial and final temperatures, and k is a constant.
In this case, we want to find the temperature (T2) at which the gases in the can will reach a pressure of 90 lbs/in^2 (P2), given that the initial temperature (T1) is 27°C and the initial pressure (P1) is 30 lbs/in^2.
First, let's convert the initial temperature from Celsius to Kelvin. We add 273.15 to the Celsius temperature:
T1 = 27°C + 273.15 = 300.15 K
Next, we plug the values into the combined gas law equation and solve for T2:
(30 lbs/in^2 * 300.15 K) / (90 lbs/in^2 * T2) = k
To solve for T2, we rearrange the equation:
T2 = (30 lbs/in^2 * 300.15 K) / (90 lbs/in^2)
Now, let's simplify the equation:
T2 ≈ (30 * 300.15) / 90
T2 ≈ 9004.5 / 90
T2 ≈ 100.05 K
Finally, to convert the temperature back to Celsius, we subtract 273.15:
T2 ≈ 100.05 K - 273.15 ≈ -173.1°C
Therefore, the temperature at which the gases in the can must be raised in order for the can to explode is approximately -173.1°C.