When a sample of carbon dioxide gas is at 36 degrees and 105 kpa is heated the new pressure of the gas becomes 165kpa. Determine the new temperature. Assume volume and amount of gas remain constant.
(P1/T1) = (P2/T2)
Remember T must be in kelvin.
To determine the new temperature, we can use the combined gas law formula:
\(\frac{{P_1 \cdot V_1}}{{T_1}} = \frac{{P_2 \cdot V_2}}{{T_2}}\)
Where:
\(P_1\) and \(P_2\) are the initial and final pressures respectively,
\(V_1\) and \(V_2\) are the initial and final volumes respectively,
and \(T_1\) and \(T_2\) are the initial and final temperatures respectively.
In this case, the volume and amount of gas remain constant, so we can remove them from the equation:
\(\frac{{P_1}}{{T_1}} = \frac{{P_2}}{{T_2}}\)
Now we can plug in the given values:
\(P_1 = 105 \, \text{kPa}\)
\(P_2 = 165 \, \text{kPa}\)
\(T_1 = 36 \, \text{°C}\) (Note: Convert to Kelvin by adding 273.15. So, \(T_1 = 309.15 \, \text{K}\))
\(T_2 = ?\) (The temperature we want to find)
Now, we rearrange the equation to solve for \(T_2\):
\(\frac{{P_2}}{{T_2}} = \frac{{P_1}}{{T_1}}\)
Next, we can multiply both sides of the equation by \(T_2\) to isolate it:
\(P_2 = \frac{{P_1}}{{T_1}} \cdot T_2\)
To solve for \(T_2\), we can divide both sides of the equation by \(\frac{{P_1}}{{T_1}}\):
\(T_2 = \frac{{P_2}}{{P_1}} \cdot T_1\)
Now we can substitute the values into the equation:
\(T_2 = \frac{{165 \, \text{kPa}}}{{105 \, \text{kPa}}} \cdot 309.15 \, \text{K}\)
Calculating this expression gives us the new temperature:
\(T_2 \approx 488.857 \, \text{K}\)
Therefore, the new temperature of the carbon dioxide gas is approximately 488.857 K.