to what temperature must a given mass of nitrogen at zero degree be heated so that both it volume and pressure will be doubled
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To determine the temperature to which the given mass of nitrogen at zero degrees Celsius must be heated so that both its volume and pressure will double, we can use a combination of the ideal gas law and the relationships between the volume, pressure, and temperature for an ideal gas.
The ideal gas law equation is:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
To find the temperature at which both the volume and pressure will be doubled, we need to consider that the volume of an ideal gas is directly proportional to its temperature, assuming other variables are constant (Boyle's Law), and the pressure of an ideal gas is also directly proportional to its temperature, again assuming other variables are constant (Gay-Lussac's Law).
So, if we double the volume and pressure, we can express the new volume (V') and pressure (P') as:
V' = 2V
P' = 2P
Substituting these expressions into the ideal gas law equation, we have:
2PV = nRT'
Where T' is the new temperature. Dividing both sides by 2P, we get:
V = (nRT') / (2P)
Now, we know that the volume is directly proportional to the temperature, so we can express the ratio of the initial volume (V) to the new temperature (T') as:
V / T' = constant
Rearranging this equation, we have:
T' = V / constant
Since the volume is being doubled, V' = 2V, so we can rewrite the equation as:
T' = (2V) / constant
Similarly, we know that the pressure is directly proportional to the temperature, so we can express the ratio of the initial pressure (P) to the new temperature (T') as:
P / T' = constant
Rearranging this equation, we have:
T' = P / constant
Since the pressure is being doubled, P' = 2P, so we can rewrite the equation as:
T' = (2P) / constant
Combining the two expressions for T', we have:
T' = (2V) / constant = (2P) / constant
Since both equations are equal to T', we can equate them to find the constant:
(2V) / constant = (2P) / constant
Simplifying further, we get:
2V = 2P
Now, since V is directly proportional to T', we can write:
V / T' = constant
Substituting the value we found for V, we have:
(2V) / T' = constant
Replacing V with its expression from the equation 2V = 2P, we get:
(2P) / T' = constant
Now, we have two equations for the constant value:
2V / constant = (2P) / constant
(2P) / constant = (2P) / constant
Simplifying, we see that both equations are equal to 1. Therefore, we know that the constant is equal to 1.
Substituting the constant value into the equation T' = (2V) / constant, we get:
T' = 2V
So, to find the temperature to which the given mass of nitrogen at zero degrees Celsius must be heated so that both its volume and pressure will double, we can simply multiply the initial temperature by 2:
T' = 2(0 degrees Celsius) = 0 degrees Celsius
Therefore, the temperature would remain zero degrees Celsius.