The vapor pressure of water at 100°C is 101 kN/m2, because water boils under these conditions. The vapor pressure of water decreases approximately linearly with decreasing temperature at a rate of 3.1 kN/m2/°C. Calculate the boiling temperature of water at an altitude of 3000 m, where the atmospheric pressure is 69 kN/m2 absolute
The Correct Answer and Explanation is:
To determine the boiling temperature of water at an altitude of 3000 m, we use the concept that water boils when its vapor pressure equals the surrounding atmospheric pressure.
At sea level, the vapor pressure of water at its boiling point (100 °C) is 101 kN/m². At higher altitudes, atmospheric pressure is lower, so water boils at a lower temperature. The vapor pressure of water decreases approximately linearly with decreasing temperature, at a rate of: dPdT=−3.1 kN/m2/°C\frac{dP}{dT} = -3.1 \, \text{kN/m}^2/\text{°C}
Step-by-step Solution:
Let’s define:
- PatmP_{\text{atm}} = atmospheric pressure at 3000 m = 69 kN/m²
- PrefP_{\text{ref}} = vapor pressure at 100 °C = 101 kN/m²
- TrefT_{\text{ref}} = 100 °C
- TbT_b = boiling temperature at 3000 m
- ΔP=Patm−Pref=69−101=−32 kN/m2\Delta P = P_{\text{atm}} – P_{\text{ref}} = 69 – 101 = -32 \, \text{kN/m}^2
Now use the linear relationship between pressure and temperature: \Delta T = \frac{\Delta P}{\frac{dP}{dT}} = \frac{-32}{-3.1} \approx 10.32\,^\circ\text{C}
So the boiling temperature: T_b = 100 – 10.32 = \boxed{89.68\,^\circ\text{C}}
Explanation
At standard atmospheric pressure (101 kN/m²), water boils at 100 °C. Boiling occurs when the vapor pressure of water equals the ambient atmospheric pressure. As altitude increases, atmospheric pressure decreases. At 3000 meters above sea level, the atmospheric pressure is only about 69 kN/m². Since water’s vapor pressure must match the surrounding pressure to boil, the boiling point must be lower at this altitude.
We are told that the vapor pressure of water decreases approximately linearly with temperature at a rate of 3.1 kN/m² per °C decrease. This means for every degree Celsius the temperature drops, the vapor pressure drops by 3.1 kN/m². By calculating how much lower the atmospheric pressure is at 3000 m compared to sea level (a difference of 32 kN/m²), we can determine the corresponding temperature decrease using this rate.
By dividing the pressure drop (32 kN/m²) by the rate (3.1 kN/m²/°C), we find that the boiling point drops by approximately 10.32 °C. Therefore, water will boil at approximately 89.68 °C at this altitude.
This lower boiling point has practical implications. For instance, cooking food (like boiling eggs or rice) takes longer at high altitudes because water cannot get as hot as it does at sea level. Engineers and scientists must consider this when designing equipment or conducting experiments involving heating or pressure at high elevations.
