A gas has a pressure of 5.7ATM at 100.0 C what is the pressure at 20.0 C? (assume voulme is unchanged)
200 C = 373 K
20 C = 293 K
P1 / T1 = P2 / T2
or
P2 = P1 T2/T1
= 5.7 (293/373)
To solve this problem, we can use the combined gas law equation, which is derived from the ideal gas law. The combined gas law equation is:
P1 * V1 / T1 = P2 * V2 / T2
Where:
P1 = Initial pressure (5.7 ATM)
V1 = Initial volume (unchanged)
T1 = Initial temperature (100.0 °C + 273.15 K = 373.15 K)
P2 = Final pressure (unknown)
V2 = Final volume (unchanged)
T2 = Final temperature (20.0 °C + 273.15 K = 293.15 K)
Now we can plug in the values into the equation and solve for P2:
(5.7 ATM) * V1 / (373.15 K) = P2 * V2 / (293.15 K)
Since the volume is unchanged, V1/V2 will cancel out, leaving us with:
(5.7 ATM) / (373.15 K) = P2 / (293.15 K)
To solve for P2, we can cross-multiply and rearrange the equation:
P2 = (5.7 ATM) * (293.15 K) / (373.15 K)
Calculating this equation gives us:
P2 = 4.47 ATM
Therefore, the pressure at 20.0 °C is approximately 4.47 ATM when the initial pressure is 5.7 ATM at 100.0 °C.
To calculate the pressure of a gas at a different temperature, we can use the ideal gas law equation:
PV = nRT
Where:
P = Pressure
V = Volume (assumed to be constant)
n = number of moles of gas
R = ideal gas constant
T = Temperature
To find the pressure at a different temperature, we need to use the ratio of the temperatures (in Kelvin) and solve for the new pressure.
Let's break down the steps:
Step 1: Convert temperatures to Kelvin
To convert Celsius to Kelvin, we add 273.15 to the Celsius value.
Given:
Temperature 1 (T1) = 100.0 °C
Temperature 2 (T2) = 20.0 °C
T1 in Kelvin = 100.0 + 273.15 = 373.15 K
T2 in Kelvin = 20.0 + 273.15 = 293.15 K
Step 2: Set up the ratio of temperatures
We need to calculate the ratio of the temperatures in Kelvin.
Temperature ratio (T2/T1) = 293.15 K / 373.15 K
Step 3: Set up the equation and solve for P2
Using the ideal gas law, we can rewrite the equation as:
P1V1/T1 = P2V2/T2
Given:
P1 = 5.7 atm (pressure at T1)
T1 = 373.15 K
V1 = constant
T2 = 293.15 K
V2 = constant (same volume as V1)
Plug in the given values into the equation:
P1/T1 = P2/T2
Solve for P2:
P2 = (P1 * T2) / T1
Now, substitute the values we have:
P2 = (5.7 atm * 293.15 K) / 373.15 K
Calculating this expression will give us the pressure at 20.0 °C.