Plot how the saturation temperature of water increases with pressure. Use the full range from the triple point to critical point
The Correct Answer and Explanation is:
Analyzed
pythonAlways show detailsCopyimport matplotlib.pyplot as plt
import numpy as np
# Define the range of pressures from the triple point to the critical point for water
# Triple point pressure: 611.657 Pa (~0.00604 atm)
# Critical point pressure: 22.064 MPa
# Pressures in Pascals
pressures = np.logspace(np.log10(611.657), np.log10(22.064e6), 500)
# Antoine equation constants for water (valid for most of the liquid-vapor range)
# log10(P) = A - (B / (C + T)), where P is in mmHg and T is in Celsius
A = 8.07131
B = 1730.63
C = 233.426
# Convert pressure from Pa to mmHg (1 atm = 101325 Pa = 760 mmHg)
pressures_mmHg = pressures * 760 / 101325
# Calculate saturation temperature in Celsius
# Rearranged Antoine equation: T = B / (A - log10(P)) - C
log_pressures = np.log10(pressures_mmHg)
temps_celsius = B / (A - log_pressures) - C
# Convert to Kelvin
temps_kelvin = temps_celsius + 273.15
# Plotting
plt.figure(figsize=(10, 6))
plt.plot(pressures / 1e6, temps_kelvin, color='blue')
plt.xlabel("Pressure (MPa)")
plt.ylabel("Saturation Temperature (K)")
plt.title("Saturation Temperature of Water vs Pressure")
plt.grid(True)
plt.tight_layout()
plt.show()
The graph above shows how the saturation temperature of water increases with pressure, ranging from the triple point to the critical point.
The triple point of water occurs at approximately 611.657 pascals, or about 0.00604 atmospheres, and the critical point occurs at about 22.064 megapascals. At the triple point, water can exist simultaneously in solid, liquid, and vapor phases. As pressure increases beyond this point, a higher temperature is required for water to boil, meaning the saturation temperature also increases.
This curve is derived using the Antoine equation, an empirical formula that relates the vapor pressure and temperature of a substance. The equation takes the form:
log₁₀(P) = A – B / (C + T)
Where:
- P is the vapor pressure in mmHg
- T is the temperature in degrees Celsius
- A, B, and C are constants specific to water
By rearranging this equation, we can solve for temperature at different pressures. As pressure increases, the logarithm of pressure increases, causing the saturation temperature to rise as well.
The curve shows a steep increase in saturation temperature at lower pressures. This means small changes in pressure at low levels result in significant changes in boiling temperature. As pressure continues to increase, the rate of temperature rise gradually slows, becoming almost flat near the critical point. At this point, the distinction between liquid and vapor disappears, and no further phase change occurs.
Understanding this relationship is important in applications like steam engines, pressure cookers, and industrial boilers, where controlling temperature and pressure is crucial for safety and efficiency.
