The Clausius-Clapeyron Law relates the vapor pressure of...

The Clausius-Clapeyron Law relates the vapor pressure of water $P$ (in atmospheres) to the temperature $T$ (in kelvins): $\frac{d P}{d T}=k \frac{P}{T^{2}}$ where $k$ is a constant. Estimate $d P / d T$ for $T=303,313,323,333,343$ using the data and the symmetric difference approximation $\frac{d P}{d T} \approx \frac{P(T+10)-P(T-10)}{20}$ \begin{tabular}{|lccccccc|} \hline$T(\mathrm{~K})$ & 293 & 303 & 313 & 323 & 333 & 343 & 353 \\ \hline$P(a \mathrm{tm})$ & 0.0278 & 0.0482 & 0.0808 & 0.1311 & 0.2067 & 0.3173 & 0.4754 \\ \hline \end{tabular} Do your estimates seem to confirm the Clausius-Clapeyron Law? What is the approximate value of $k$ ?

The Clausius-Clapeyron Law relates the vapor pressure of water $P$ (in atmospheres) to the temperature $T$ (in kelvins):
$\frac{d P}{d T}=k \frac{P}{T^{2}}$
where $k$ is a constant. Estimate $d P / d T$ for $T=303,313,323,333,343$ using the data and the symmetric difference approximation
$\frac{d P}{d T} \approx \frac{P(T+10)-P(T-10)}{20}$
\begin{tabular}{|lccccccc|}
\hline$T(\mathrm{~K})$ & 293 & 303 & 313 & 323 & 333 & 343 & 353 \\
\hline$P(a \mathrm{tm})$ & 0.0278 & 0.0482 & 0.0808 & 0.1311 & 0.2067 & 0.3173 & 0.4754 \\
\hline
\end{tabular}
Do your estimates seem to confirm the Clausius-Clapeyron Law? What is the approximate value of $k$ ?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Finite Difference Methods

Finite difference methods are numerical techniques used to approximate derivatives by using function values at discrete points. The symmetric difference approximation, in particular, uses values from both sides of the point of interest, which helps to cancel out certain error terms and often provides a more accurate estimate of the derivative than one-sided methods.

Numerical Verification of Theoretical Models

Numerical verification involves using experimental or numerical data to test the validity of theoretical laws. By applying concepts such as the symmetric difference approximation to data, one can estimate derivatives and compare them against theoretical predictions, such as those provided by the Clausius-Clapeyron Law, to determine if the observed behavior conforms to the expected mathematical models.

Clausius-Clapeyron Law

The Clausius-Clapeyron Law is a thermodynamic relation that describes how the vapor pressure of a substance depends on temperature. It is derived from considerations of phase equilibrium and latent heat, showing that the rate of change of pressure with respect to temperature is proportional to the vapor pressure divided by the square of the temperature. This law is essential for understanding phase transitions such as evaporation and condensation.

Transcript

Please give Ace some feedback