Show that the temperature measured on the Kelvin scale agree with these measured on the perfect gas scale.
The Kelvin temperature, denoted as \(T_K\), is related to the Celsius temperature \(T_C\) by the equation:
\[ T_K = T_C + 273.15 \]
Now, let's consider the perfect gas law, which relates the pressure \(P\), volume \(V\), and temperature \(T\) of a gas:
\[ PV = nRT \]
Where:
\( P \) = Pressure
\( V \) = Volume
\( n \) = Number of moles of gas
\( R \) = Ideal gas constant
\( T \) = Temperature (in Kelvin)
Since the Kelvin scale is based on the behavior of ideal gases, the temperature measured in Kelvin is directly related to the kinetic energy of gas molecules. This makes it consistent with the behavior of gases as described by the ideal gas law.
So, the temperature measured on the Kelvin scale agrees with that measured on the perfect gas scale because both are based on the same underlying principles of ideal gas behavior. The Kelvin scale provides an absolute temperature measurement that directly reflects the kinetic energy of gas molecules, and this aligns with the ideal gas law.
Suppose we have a certain amount of gas in a container with a volume of 1.00 L. The pressure is 2.00 atm. We want to find the temperature of the gas in Kelvin using the ideal gas law.
Given:
\( P = 2.00 \, \text{atm} \)
\( V = 1.00 \, \text{L} \)
\( R = 0.0821 \, \text{L atm / K mol} \)
We'll use the ideal gas law equation \( PV = nRT \) to solve for the temperature \( T \):
\[ T = \frac{PV}{nR} \]
Since the number of moles \( n \) is not given, we'll assume \( n = 1 \) for simplicity.
Substituting the given values:
\[ T = \frac{(2.00 \, \text{atm})(1.00 \, \text{L})}{(1)(0.0821 \, \text{L atm / K mol})} \]
Calculating:
\[ T \approx 24.39 \, \text{K} \]
So, the temperature of the gas in Kelvin is approximately 24.39 K.
This example demonstrates how the Kelvin temperature, based on the ideal gas law, agrees with the behavior of gases as described by the perfect gas scale. It showcases the consistent relationship between the two scales and how temperature measurements made on one scale are directly applicable to the other.
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