Suppose \(35.0 \mathrm{~g}\) of steam at \(100 .^{\circ} \mathrm{C}\) are added to \(300 \mathrm{~g}\) of water at \(25^{\circ} \mathrm{C}\). Is this sufficient steam to heat all the water to \(100 .^{\circ} \mathrm{C}\) and still have steam remaining? Show evidence for your answer.

The specific heat capacity is a crucial concept in thermal dynamics. It measures how much heat energy is needed to raise the temperature of a given mass of a substance by 1 degree Celsius. The formula to calculate the heat required is: \( q = mc\Delta T \) Here:

• \( q \) is the heat energy (in joules)
• \( m \) is the mass of the substance (in grams)
• \( c \) is the specific heat capacity (in J/g°C)
• \( \Delta T \) is the change in temperature (in °C)

For example, in the given exercise, you need to calculate the heat needed to raise \( 300 \, g \) of water from \( 25^{\circ} C \) to \( 100^{\circ} C \). Knowing that the specific heat capacity of water is \( 4.18 \, J/g^{\circ} C \), you multiply the mass, specific heat capacity, and the temperature change: \[ q_{water} = 300 \, g \times 4.18 \, J/g^{\circ} C \times (100^{\circ} C - 25^{\circ} C) = 300 \, g \times 4.18 \, J/g^{\circ} C \times 75^{\circ} C = 94050 \, J \] Thus, we'd require \( 94050 \, J \) of energy to raise the temperature of the given water.

Latent heat of vaporization is another critical concept in thermodynamics associated with phase changes. It refers to the amount of heat required to change one gram of a substance from liquid to gas or vice versa, at its boiling point, without changing its temperature. The formula to calculate this energy is: \ q = mL \ Here:

• \( q \) is the heat energy (in joules)
• \( m \) is the mass (in grams)
• \( L \) is the latent heat of vaporization (in J/g)

In the exercise, the latent heat of vaporization of water is \( 2260 \, J/g \). To find the heat released by the condensation of \( 35 \, g \) of steam, you multiply the mass by the latent heat: \[ q_{steam} = 35 \, g \times 2260 \, J/g = 79100 \, J \] Thus, condensing \( 35 \, g \) of steam releases \( 79100 \, J \) of energy.

Energy change calculation combines both specific heat capacity and latent heat of vaporization concepts to solve problems involving temperature changes and phase transitions. To determine if the steam can sufficiently heat the water to \( 100^{\circ} C \), follow these steps:

• Calculate the heat required to raise the water temperature to \( 100^{\circ} C \): \[ q_{water} = 300 \, g \times 4.18 \, J/g^{\circ} C \times 75^{\circ} C = 94050 \, J \]
• Calculate the heat released by steam condensation: \[ q_{steam} = 35 \, g \times 2260 \, J/g = 79100 \, J \]
• Compare the two values. The heat required (\( 94050 \, J \)) is more than the heat released (\( 79100 \, J \)).

Since the heat provided by steam is insufficient, it indicates that not enough steam is present to heat all the water to \( 100^{\circ} C \). No steam will remain once the water reaches partial heating.