The pressure of a gas varies directly as its temperature. IF the pressure of a gas is 200 pounds per square inch (lb/in^2) at a temperature of 65.5 degrees C, what is the pressure of the gas when the temperature is 250 degrees F?
The pressure of a gas varies directly as its temperature:
P = kT
Note that the temperature must be absolute (in Kelvin units). To convert, just add 273 to the degree Celsius.
Substituting the given,
200 lb/in^2 = k (65.5 + 273)
k = 200 / 338.5
k = 0.5908
Now that we know the k, we can use this to solve for the pressure in the second condition. Note that the temperature is in degree F. To convert, T,K = (5/9)*(T,F + 459.7).
P = 0.5908 * (5/9) * (250 + 459.7)
P = 232.94 lb/in^2
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To find the pressure of the gas when the temperature is 250 degrees Fahrenheit, we need to use the direct variation relationship between pressure and temperature. However, we need to convert the given temperature from Fahrenheit to Celsius before proceeding.
1. Convert 250 degrees Fahrenheit to Celsius:
To convert Fahrenheit to Celsius, you can use the formula:
Celsius = (Fahrenheit - 32) * 5/9
Calculating:
Celsius = (250 - 32) * 5/9
= 218 * 5/9
= 120.56 degrees Celsius (rounded to two decimal places)
2. Now that we have the temperature in Celsius, we can use the direct variation relationship between pressure and temperature to find the pressure of the gas.
According to the problem, the pressure varies directly as the temperature. This means we can write the relationship as:
P = kT
Where P is the pressure, k is the constant of variation, and T is the temperature in degrees Celsius.
3. Substitute the initial pressure and temperature values into the equation to find the constant of variation (k).
P = kT
200 = k * 65.5
Solve for k:
k = 200 / 65.5
≈ 3.05
4. Now that we have the value of k, we can use the equation to find the pressure (P) when the temperature (T) is 120.56 degrees Celsius (from step 1).
P = kT
= 3.05 * 120.56
≈ 368.16 pounds per square inch (rounded to two decimal places)
Therefore, the pressure of the gas when the temperature is 250 degrees Fahrenheit is approximately 368.16 pounds per square inch (lb/in^2).
To solve this problem, we need to use the direct variation formula that relates pressure and temperature. The formula states that:
Pressure = k * Temperature
Where "k" is a constant of variation. This means that if the temperature (in this case) increases, the pressure will also increase proportionally.
To find the constant of variation, we can use the information given in the problem. We are told that the pressure is 200 pounds per square inch (lb/in^2) at a temperature of 65.5 degrees Celsius. Let's convert this temperature to Fahrenheit:
65.5 degrees Celsius * 9/5 + 32 = 149.9 degrees Fahrenheit
Now, we can use the given temperature and pressure to find the constant of variation (k):
200 lb/in^2 = k * 149.9 degrees Fahrenheit
Divide both sides of the equation by 149.9 degrees Fahrenheit:
k = 200 lb/in^2 / 149.9 degrees Fahrenheit
k ≈ 1.334 lb/in^2/degree Fahrenheit (rounded to three decimal places)
Now that we have the constant of variation (k), we can use it to find the pressure (P) when the temperature is 250 degrees Fahrenheit.
P = k * Temperature
P = 1.334 lb/in^2/degree Fahrenheit * 250 degrees Fahrenheit
P ≈ 333.5 lb/in^2 (rounded to one decimal place)
Therefore, the pressure of the gas when the temperature is 250 degrees Fahrenheit is approximately 333.5 pounds per square inch (lb/in^2).