Question

In Denver, the Mile-High City, water boils at \(95^{\circ} \mathrm{C}\). What is atmospheric pressure in atmospheres in Denver? \(\Delta H_{\text {vap }}\) for \(\mathrm{H}_{2} \mathrm{O}\) is \(40.67 \mathrm{~kJ} / \mathrm{mol}\)

Short answer

The atmospheric pressure in Denver is approximately 0.947 atm.

Step-by-step solution

Understand the Problem

To solve the problem, we need to determine the atmospheric pressure in Denver using the boiling point of water, which is given as \(95^{\circ} \text{C}\). We'll use the Clausius-Clapeyron equation, which relates pressure and temperature to the enthalpy of vaporization.

Set Up the Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is: \(\ln \left(\frac{P_2}{P_1}\right) = \frac{\Delta H_{\text{vap}}}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)\)where \(\Delta H_{\text{vap}}\) is the enthalpy of vaporization, \(R\) is the ideal gas constant (8.314 J/mol·K), and \(T_1\) and \(T_2\) are absolute temperatures in Kelvin. \(P_1\) is 1 atm, the standard boiling point pressure.

Convert Temperatures to Kelvin

Convert the given boiling point of water in Denver from Celsius to Kelvin:\(T_2 = 95^{\circ}C + 273.15 = 368.15\, K\)And use the standard boiling point of water:\(T_1 = 100^{\circ}C + 273.15 = 373.15\, K\)

Rearrange and Solve for \(P_2\)

Rearrange the Clausius-Clapeyron equation to solve for \(P_2\):\(P_2 = P_1 \times \exp\left(\frac{\Delta H_{\text{vap}}}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)\right)\)Substitute the values:\(P_2 = 1 \times \exp\left(\frac{40.67 \times 10^3}{8.314} \left(\frac{1}{373.15} - \frac{1}{368.15}\right)\right)\)

Calculate the Pressure

Calculate the exponential part:\(P_2 = \exp\left(\frac{40,670}{8.314} \times (0.00268 - 0.00272)\right)\)Calculate:\(P_2 = \exp(-0.0542) \approx 0.947\, \text{atm}\)

Key concepts

Enthalpy of Vaporization

When we talk about the enthalpy of vaporization, we're discussing the energy required for a liquid to change into a vapor. Specifically, this concept refers to the heat needed to transform one mole of a liquid into gas at constant temperature and pressure. This energy change is vital for processes like boiling, where heat is applied to water, raising it from a liquid to a vapor state.

• Measured in ext{J/mol} or ext{kJ/mol}, it indicates the robustness of intermolecular forces within a liquid.
• For water, ext{ΔH_{vap}} is typically around 40.67 ext{kJ/mol}, making it relatively high due to strong hydrogen bonds.
• This value tells us just how much energy is required to overcome these bonds for vaporization to occur.
• The more significant the enthalpy of vaporization, the more robust the substance's intermolecular forces.

Understanding ext{ΔH_{vap}} is not just about knowing a number; it is about comprehending the energy dynamics within liquids.

Atmospheric Pressure

Atmospheric pressure represents the force exerted by air molecules pressing down on Earth's surface. It is often expressed in units such as atmospheres (atm) or Pascals (Pa). Atmospheric pressure varies with altitude, which is why locations like Denver, situated at a higher altitude, experience different pressure levels than areas at sea level.

• At sea level, standard atmospheric pressure is typically 1 atm.
• In Denver, the pressure is lower; this decrease is due to the reduced weight of the atmosphere above.
• Pressure affects how substances boil; less atmospheric pressure means water can boil at lower temperatures.

The connection between atmospheric pressure and boiling point is crucial to understanding how changes in altitude affect everyday things like cooking or even physiological processes.

Boiling Point of Water

The boiling point is the temperature at which a liquid turns into vapor. For water, it's commonly known as 100°C at sea level. However, this isn't universal. Factors like atmospheric pressure influence the boiling point, leading to notable variations.

• At higher altitudes like in Denver, reduced pressure means water boils at about 95°C instead of 100°C.
• This adjustment is because boiling occurs when vapor pressure equals atmospheric pressure.
• A lower atmospheric pressure results in a lower vapor pressure required to initiate boiling.

Knowing the boiling point at different pressures is essential for practical applications, such as cooking or scientific experiments, ensuring accurate predictions and outcomes.