You now have the bacteria needed to create the fuel. "Computer, what is the next step in this process?" Your computer responds that the bacteria have given off enough gas (collected in a 18

Question

L closed container) to create the fuel, and that the fuel is developed by increasing the temperature of the gas at a rate such that the pressure will initially rise at a rate of 41.5725
kPa/min (kilopascals per minute). You control the initial rate of change of temperature, and must set this value so that the initial rate of change of pressure is 41.5725
kPa/min. Once you have made this initial adjustment, the system will adjust dynamically until the fuel is created.



Your computer informs you that this gas follows the Ideal Gas Law ( PV=nRT
), and that the units for these calculations are already accounted for. This means that no conversions are necessary in your calculations. Some initial values (at time t=0
) are provided below:

Answer 1

Great! Let's solve this step-by-step using the Ideal Gas Law.

The Ideal Gas Law is given by:
\[ PV = nRT \]

Where:
- \( P \) is the pressure
- \( V \) is the volume (18 L in this case)
- \( n \) is the amount of substance (number of moles)
- \( R \) is the ideal gas constant
- \( T \) is the temperature in Kelvin

Given information:
- Volume \( V = 18 \) L
- Ideal Gas Constant \( R \)
- Initial rate of change of pressure \( \frac{dP}{dt} = 41.5725 \) kPa/min

The rate of change of pressure with respect to time can be related to the rate of change of temperature with respect to time if we consider the relationship derived from the Ideal Gas Law:

Looking at the Ideal Gas Law:
\[ P = \frac{nRT}{V} \]

Taking the derivative with respect to time \( t \):
\[ \frac{dP}{dt} = \frac{nR}{V} \frac{dT}{dt} \]

Given:
\[ \frac{dP}{dt} = 41.5725 \text{ kPa/min} \]

We can rearrange to solve for \( \frac{dT}{dt} \):
\[ \frac{dT}{dt} = \frac{\left(\frac{dP}{dt}\right) \times V}{nR} \]

Let’s denote the initial values from the ideal gas law to solve \( n \) or other remaining required information.
If you have the value for \( R \) and the initial conditions (initial pressure \( P \) and initial temperature \( T \)), we can proceed.

Given that the units are accounted for:
- \( R \approx 8.314 \) J/(mol·K) converted suitably for our scenario

If you have \( P \) during \( t=0 \) and the initial \( T \):
\[ n = \frac{PV}{RT} \]

Once you have \( n \) and \( R \), you can easily calculate:
\[ \frac{dT}{dt} = \frac{(41.5725\ \text{kPa/min} \times 18\ \text{L})}{n \times R} \]

However, if the exact initial temperature and pressure values are missing here, you would ideally plug in and use them for determining moles \( n \) and further solve accordingly.

Could you please confirm the initial pressure \( P \) and initial temperature \( T \) values so that we finalize the rate of change of temperature?