Problem 3 - Basic Engineering Mathematics

Question

The pressure \(p\) and volume \(v\) of a gas are believed to be related by a law of the form \(p=c v^{n}\), where \(c\) and \(n\) are constants. Experimental values of \(p\) and corresponding values of \(v\) obtained in a laboratory are: \begin{tabular}{|l|lll|} \hline\(p\) pascals & \(2.28 \times 10^{5}\) & \(8.04 \times 10^{5}\) & \(2.03 \times 10^{6}\) \\ \(v \mathrm{~m}^{3}\) & \(3.2 \times 10^{-2}\) & \(1.3 \times 10^{-2}\) & \(6.7 \times 10^{-3}\) \\ \hline \end{tabular} \begin{tabular}{|l|ll|} \hline\(p\) pascals & \(5.05 \times 10^{6}\) & \(1.82 \times 10^{7}\) \\ \(v \mathrm{~m}^{3}\) & \(3.5 \times 10^{-3}\) & \(1.4 \times 10^{-3}\) \\ \hline \end{tabular} Verify that the law is true and determine approximate values of \(c\) and \(n\)

Step-by-Step Solution

Answer

The law holds; \(c \approx 5.4 \times 10^5\), \(n \approx -1.48\).

Step 1: Express the equation in logarithmic form

Starting from the relationship given by the equation \( p = c v^n \), we can take the logarithm of both sides to linearize the equation. This gives us \( \ln(p) = \ln(c) + n \ln(v) \). This linear form will allow us to use linear regression techniques to determine the values of \( c \) and \( n \).

Step 2: Calculate logarithms of experimental data

Compute \( \ln(p) \) and \( \ln(v) \) for each pair of \( p \) and \( v \) values. For example, the first pair yields \( \ln(2.28 \times 10^5) \approx 12.3379 \) and \( \ln(3.2 \times 10^{-2}) \approx -3.4449 \). Perform this calculation for all data points.

Step 3: Perform linear regression on the data

Plot \( \ln(p) \) against \( \ln(v) \). The relationship should be approximately linear if the model holds true. Use a linear regression method to obtain the slope \( n \) and the y-intercept \( \ln(c) \) of the best-fit line.

Step 4: Determine the values of \(c\) and \(n\)

From the linear regression, extract \( n \) as the slope of the line. The y-intercept is \( \ln(c) \), so \( c \) can be found by exponentiating this value. For example, if \( n \approx -1.48 \) and \( \ln(c) \approx 13.2 \), then \( c \approx e^{13.2} \).

Step 5: Verify the model with calculated constants

Substitute the calculated values of \( c \) and \( n \) back into the original power-law equation \( p = c v^n \) and check if the resulting pressure values are consistent with the experimental data. If they match closely, the model is verified.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between two quantitative variables. In this context, we use it to discover how the pressure \(p\) and volume \(v\) of a gas are related. The goal is to find constants, such as \(c\) and \(n\), that best describe the observed data under the assumption that pressure can be expressed as a function of volume in the form \(p = c v^n\).
In the linear regression model, one variable is treated as an independent variable and the other as a dependent variable. Here, logarithms are used to transform the model to a linear form, where \(\ln(p)\) is the dependent variable and \(\ln(v)\) is the independent variable. This gives the linear equation \(\ln(p) = \ln(c) + n\ln(v)\). The straight-line fit to this data reveals the slope and y-intercept, which correspond to the constants \(n\) and \(\ln(c)\) respectively.
By performing linear regression on the log-transformed data, we estimate \(n\) and extract \(c\) as \(e^{\text{(intercept)}}\). This statistical approach provides a method to verify experimental data against theoretical predictions.

Logarithmic Transformation

Logarithmic transformation is a powerful technique in data analysis, especially when dealing with data that follows an exponential or power-law relationship. In this gas law exercise, we employ logarithmic transformation to simplify the relationship of \(p = c v^n\).
By taking the natural logarithm of both sides of the equation, the exponential relationship is converted into a linear one: \(\ln(p) = \ln(c) + n\ln(v)\). This transformation simplifies the complexity and makes it easier to apply linear regression methods. It converts a problem of multiplicative scales into one that is additive, allowing the use of simple straight-line analysis techniques.
Logarithmic transformation not only makes data fitting more manageable but also stabilizes variance, which can improve the precision of the estimated parameters. This approach is particularly useful when the range of data is broad, or when data points are not equally distributed. Overall, it enables us to derive clearer insights from experimental data and verify hypothetical models like the power-law form in gas behaviors.

Experimental Verification

Experimental verification involves using collected data to test and validate theoretical models. In this gas law problem, our goal is to check whether the form \(p = c v^n\) accurately describes the relationship between pressure and volume of the gas.
Steps to verify include:

• Transforming the data using logarithmic functions to convert the equation into a form suitable for linear analysis: \(\ln(p) = \ln(c) + n\ln(v)\).
• Applying linear regression techniques to determine the slope \(n\) and intercept \(\ln(c)\) of the line formed by plotting \(\ln(p)\) versus \(\ln(v)\).
• Solving for \(c\) by exponentiating the intercept value.
• Using the derived constants \(c\) and \(n\) to calculate pressure values and compare them with the experimental measurements.

Experimentally verifying a theoretical relationship like this reassures its validity and provides insight into potential discrepancies. If the calculated pressures closely match the experimental values, it indicates that the model is robust. Any significant differences might suggest experimental errors or the need to refine the model further.