Question

Plot the viscosity of water. The viscosity of water, \(\mu\), varies with the temperature \(T\) (in Kelvin) according to $$ \mu(T)=A \cdot 10^{B /(T-C)} $$ where \(A=2.414 \cdot 10^{-5} \mathrm{~Pa} \mathrm{~s}, B=247.8 \mathrm{~K}\), and \(C=140 \mathrm{~K}\). Plot \(\mu(T)\) for \(T\) between 0 and 100 degrees Celsius. Label the \(x\) axis with 'temperature (C)' and the \(y\) axis with 'viscosity (Pa s)'. Note that \(T\) in the formula for \(\mu\) must be in Kelvin. Name of program file: water_viscosity.py.

Short answer

Plot of water viscosity against temperature is implemented using Python, showing how viscosity decreases as temperature rises.

Step-by-step solution

Understanding the Formula

The given formula for the viscosity of water, \( \mu(T) = A \cdot 10^{B/(T-C)} \), requires constants \( A = 2.414 \times 10^{-5} \, \mathrm{Pa\, s}, B = 247.8 \, \mathrm{K} \), and \( C = 140 \, \mathrm{K} \). Temperature \( T \) is in Kelvin.

Temperature Conversion

Given the range is 0 to 100 degrees Celsius, convert this to Kelvin by adding 273.15 to each Celsius value. Thus, the range in Kelvin is from \( 273.15 \, \mathrm{K} \) to \( 373.15 \, \mathrm{K} \).

Implementing the Formula in Code

Create a Python file named `water_viscosity.py`. Use a Python library like Matplotlib for plotting, and NumPy for calculations. Define the required constants and write a function to compute \( \mu(T) \) over the temperature range.

Creating the Plot

Use Matplotlib to plot the graph of viscosity against temperature. Set the x-axis (temperature) range from 0 to 100 degrees Celsius and label it. The y-axis should represent viscosity in Pascal-seconds (Pa s), labeled accordingly.

Verifying and Saving the Plot

Ensure the plot is correct by verifying the expected behavior of the plot (viscosity decreases with increased temperature). Save the plot as an image file if needed using Matplotlib's `savefig` function.

Key concepts

Viscosity of Water Calculation

To calculate the viscosity of water, we use the formula: \[ \mu(T) = A \cdot 10^{B/(T-C)} \] Here, the constants are given as follows: - \( A = 2.414 \times 10^{-5} \) Pa s, - \( B = 247.8 \) K, - \( C = 140 \) K. In this formula, \( T \) represents the temperature in Kelvin. The goal is to calculate \( \mu \), the viscosity, as temperature changes between the given limits. Viscosity tends to decrease as temperature increases. This is due to the molecular interactions becoming less significant at higher temperatures. This behavior is important in many scientific and engineering applications. Understanding and calculating viscosity is essential for studying fluid dynamics and designing systems involving fluid flow. Using the given formula, we need to compute \( \mu(T) \) for the specific range of temperatures defined for this problem.

Temperature Conversion in Python

When dealing with scientific calculations, it is important to use consistent units. This often means converting temperatures from Celsius to Kelvin. In our exercise, we are working with temperatures from 0 to 100 degrees Celsius. To convert these to Kelvin, we use the simple formula: \[ T_{\text{Kelvin}} = T_{\text{Celsius}} + 273.15 \] This conversion is straightforward in Python. To achieve this in code, you can use NumPy to easily handle arrays of temperatures. Define an array for Celsius temperatures and convert each element to Kelvin by adding 273.15. Here's a small snippet demonstrating how you can accomplish this: ```python import numpy as np celsius_temps = np.arange(0, 101) kelvin_temps = celsius_temps + 273.15 ``` With the temperatures converted to Kelvin, you can now proceed with calculating the viscosity.

Matplotlib Plotting

Once you have the viscosity values calculated over your range of temperatures, visualizing this data is made straightforward with Matplotlib. Matplotlib is a powerful Python library that helps you create a wide array of static, interactive, and animated plots. For the given problem, you are required to plot the viscosity \(\mu(T)\) against temperature in Celsius. First, ensure you've installed Matplotlib using pip if you haven't already: ```bash pip install matplotlib ``` In your Python script, you can begin by importing Matplotlib and plotting your data: ```python import matplotlib.pyplot as plt plt.plot(celsius_temps, viscosity_values, label='Viscosity vs Temperature') plt.xlabel('Temperature (C)') plt.ylabel('Viscosity (Pa s)') plt.title('Viscosity of Water as a Function of Temperature') plt.legend() plt.grid(True) plt.show() ``` This code sets up a line plot and labels your axes and title properly. It's a good idea to include a grid and a legend for better readability of your plot. If needed, you can save your plot as an image using Matplotlib's `savefig` method.

NumPy for Scientific Calculations

In scientific computing tasks, managing large sets of data efficiently is crucial. NumPy is a highly-regarded library in Python that is built for handling large arrays and matrices, along with a variety of mathematical functions. When calculating something as detailed as the viscosity of water over a temperature range, NumPy simplifies operations like this. NumPy offers many advantages:

• It supports element-wise calculations, making operations on large datasets effortless.
• Its array objects are optimized for performance, ensuring your programs run efficiently.
• It provides a comprehensive collection of mathematical functions, perfect for scientific applications.

In our case, we use NumPy to handle an array of temperature values in Kelvin. We then utilize these values in the viscosity formula. Here's an example illustrating how this might look in Python: ```python viscosity_values = A * 10 ** (B / (kelvin_temps - C)) ``` This calculated array, `viscosity_values`, can easily be plotted using Matplotlib. NumPy's power and ease of use make it irreplaceable for performing scientific calculations like these.