Calculating Pressure - to Integrate or Not?

• erok81
In summary, the conversation discusses using the Clausius-Clapeyron relation to calculate the pressure needed to melt an ice cube at -1°C. The two methods mentioned are using ΔP/ΔT or integration, with the latter being the correct approach. It is necessary to integrate when the integrand changes significantly between the limits of integration. In this case, the integrand is 1/T which does not change much between 0°C and -1°C, making the integration method more accurate.
erok81

Homework Statement

This was a multi-part problem using the Clausius-Clapeyron relation to calculate how much pressure needed to be put on an ice cube in order to have it melt at -1*C.

Homework Equations

Clausius-Clapeyron relation given by:

$\frac{dP}{dT}=\frac{L}{T\Delta V}$

The Attempt at a Solution

I am confused on how exactly to solve this. I've seen it two ways. One integrates and one doesn't, leading to two different answers.

The first is use just using ΔP/ΔT as just changes. So...

$\frac{\Delta P}{\Delta T}=\frac{L}{T \Delta V}$

That's really it. Then one plugs in 272k for temp, then the value for ΔV that was calculated earlier, and L. Then just solve. I don't think this is the correct way.

The second method is integration.

$\frac{dP}{dT}=\frac{L}{T\Delta V}$

$\int ^{P_{f}}_{P{_i}} dP= \frac{L}{\Delta V} \int ^{T_{f}}_{T{_i}}\frac{dT}{T}$

Which, after adding Pi to both sides, the result is:

$P_{f}=\frac{L}{\Delta V} ln(\frac{T_f}{T_i})+P_i$

And then plug in the same values, using the initial P and T as STP.

So my question is twofold, is the integration method correct? And this problem isn't the best example, but when is it needed to integrate vs. only use the Δ values?

erok81 said:
The second method is integration.

$\frac{dP}{dT}=\frac{L}{T\Delta V}$

$\int ^{P_{f}}_{P{_i}} dP= \frac{L}{\Delta V} \int ^{T_{f}}_{T{_i}}\frac{dT}{T}$
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.
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So my question is twofold, is the integration method correct?
Since the Clausius-Clapeyron relation involves dP/dT and not ΔP/ΔT, then integration is the correct method, strictly speaking.

And this problem isn't the best example, but when is it needed to integrate vs. only use the Δ values?
If you examine the integral you did, the question to ask is, "does the integrand change by very much in between the two limits of integration?"

The integrand is 1/T in the integral $\int ^{T_{f}}_{T{_i}}\frac{dT}{T}$. What is the percentage change in 1/T, in going from 0°C to -1°C? Is that a large or a small change?

1. What is pressure and why is it important to calculate?

Pressure is defined as the force per unit area. It is important to calculate because it allows us to understand the behavior of gases and liquids, and how they interact with their surroundings. This information is crucial in various fields such as engineering, meteorology, and chemistry.

2. How is pressure calculated?

Pressure can be calculated using the formula: pressure = force / area. The SI unit for pressure is Pascal (Pa), which is equal to 1 N/m². Other common units for pressure include atmospheres (atm), millimeters of mercury (mmHg), and pounds per square inch (psi).

3. When should we use integration to calculate pressure?

Integration should be used to calculate pressure when dealing with non-uniform or changing pressure distributions. This is because integration allows us to take into account the varying force and area values over a given area or volume, providing a more accurate result.

4. Are there any situations where integration is not necessary when calculating pressure?

Yes, there are situations where integration may not be necessary. If the pressure distribution is uniform and there are no changes in force or area, then the pressure can be calculated using the basic formula pressure = force / area.

5. How can we apply the concept of pressure calculation in real-life scenarios?

The concept of pressure calculation is widely used in various real-life scenarios. For example, in weather forecasting, pressure calculations help predict changes in atmospheric pressure which can indicate upcoming weather patterns. In engineering, pressure calculations are used to determine the strength and stability of structures. In healthcare, pressure calculations are crucial in understanding blood pressure and respiratory functions.

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