An aerosol can contains gases under a pressure
of 4.6 atm at 18◦C. If the can is left on a hot sandy beach, the pressure of the gases increases to 4.69 atm. What is the Celsius temperature on the beach
since PV/T is constant, if P increases by a factor of 4.69/4.6, T decreases by a factor of 4.6/4.69. T is Kelvin, of course.
To find the Celsius temperature on the beach, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure (in atm)
V = volume (in liters)
n = number of moles
R = ideal gas constant (0.0821 L.atm/mol.K)
T = temperature (in Kelvin)
First, let's convert the initial pressure of 4.6 atm to Kelvin using the formula:
T = 18 + 273.15
T = 291.15 K
Next, we can calculate the final temperature (T2) on the beach using the formula:
(P1 x V1) / T1 = (P2 x V2) / T2
Let's assume that the volume of the aerosol can remains constant, so V1 = V2. Therefore, we can simplify the equation to:
P1 / T1 = P2 / T2
Now we can plug in the given values:
4.6 / 291.15 = 4.69 / T2
Cross-multiplying:
4.6 x T2 = 4.69 x 291.15
Dividing both sides by 4.6:
T2 = (4.69 x 291.15) / 4.6
T2 ≈ 297.51 K
To convert this temperature back to Celsius, subtract 273.15:
T2 ≈ 24.36°C
Therefore, the Celsius temperature on the beach is approximately 24.36°C.
To find the Celsius temperature on the beach, we need to use the ideal gas law equation, which states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature in Kelvin
First, we need to convert the temperatures from Celsius to Kelvin. The conversion formula is:
K = °C + 273.15
Given that the initial pressure (P1) is 4.6 atm and the initial temperature (T1) is 18°C, we can convert T1 to Kelvin by adding 273.15:
T1 = 18 + 273.15 = 291.15 K
Next, we are given the final pressure (P2), which is 4.69 atm. We want to find the final temperature (T2).
Now, rearranging the ideal gas law equation to solve for temperature:
T = (PV) / (nR)
Since the number of moles (n) and the volume (V) are constant, we can rewrite the equation as:
T1 = (P1 * V) / (n * R)
T2 = (P2 * V) / (n * R)
As mentioned earlier, the volume and number of moles remain constant, so we can eliminate them from the equation. Thus, the equation becomes:
T1 / P1 = T2 / P2
Now we can substitute the values into the equation:
T1 / P1 = T2 / P2
291.15 / 4.6 = T2 / 4.69
Now we can calculate T2:
T2 = (291.15 * 4.69) / 4.6
By calculating this expression, we find that T2 is approximately 297.957 K.
To convert this value back to Celsius, we subtract 273.15:
T2 = 297.957 - 273.15 = 24.807°C
Therefore, the Celsius temperature on the beach is approximately 24.807°C.