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Chapter 8: Problem 1213

In a container of negligible heat capacity, \(200 \mathrm{~g}\) ice at \(0^{\circ} \mathrm{C}\) and \(100 \mathrm{~g}\) steam at \(100^{\circ} \mathrm{C}\) are added to \(200 \mathrm{~g}\) of water that has temperature $55^{\circ} \mathrm{C}$. Assume no heat is lost to the surroundings and the pressure in the container is constant \(1 \mathrm{~atm} .\) Amount of the Sm left in the system, is equal to (A) \(16.7 \mathrm{~g}\) (B) \(8.4 \mathrm{~g}\) (C) \(12 \mathrm{~g}\) (D) \(0 \mathrm{~g}\) Copyright () StemEZ.com. All rights reserved.

Short Answer

Expert verified
The closest answer, which may have some approximation error, is \(8.4\,\text{g}\) (Option B).

Step by step solution

01

Calculate heat absorbed by the ice during melting

First, let's calculate the amount of heat required by the 200 g of ice at 0°C to completely melt into water. The heat required for melting can be calculated using the formula: \(q = mL\) where: q = heat required (Joules) m = mass of ice (grams) L = heat of fusion for ice (334 J/g) So, for our situation with 200 g of ice: \(q_{melt} = (200\,\text{g}) (334\,\text{J/g}) = 66,800\,\text{J}\)
02

Determine the equation for total heat exchange

The total heat exchange can be represented as: \(q_{melt} + q_{ice\rightarrow water} = q_{steam\rightarrow water} + q_{water}\) Now, we can calculate the heat required to raise the temperature of water (after melting) to the equilibrium temperature, and the heat lost by steam and water to reach that equilibrium temperature.
03

Calculate the heat absorbed by the melted ice

Let's denote equilibrium temperature as T. To calculate the heat absorbed by the previously melted ice converted to water, we use the formula: \(q = mc(T_{f} - T_{i})\) where: q = heat absorbed (Joules) m = mass (grams) c = specific heat of water (4.18 J/g°C) \(T_{f}\) = final temperature (°C) \(T_{i}\) = initial temperature (°C) So, for the 200 g of melted ice: \(q_{ice\rightarrow water} = (200\,\text{g}) (4.18\,\text{J/g°C}) (T - 0)\) \(q_{ice\rightarrow water} = 836T\,\text{J}\)
04

Calculate the heat lost by steam to become water

Next, we will calculate the heat lost by the 100 g of steam to become water using the formula: \(q = mL\) where: L = heat of vaporization for water (2260 J/g) So, for 100 g of steam: \(q_{steam\rightarrow water} = (100\,\text{g}) (2260\,\text{J/g}) = 226,000\,\text{J}\)
05

Calculate the heat lost by water after steam condensation

Lastly, we will calculate the heat lost by water (steam after condensation and given 200 g of water) to achieve equilibrium temperature: \(q_{water} = mc(T_{i} - T_{f})\) Since mass = 100 g steam mass + 200 g water mass = 300 g \(q_{water} = (300\,\text{g}) (4.18\,\text{J/g°C}) (55 - T)\) \(q_{water} = 1254(55 - T)\,\text{J}\)
06

Solve for equilibrium temperature (T)

Now, let's substitute our results into the heat exchange equation from Step 2: \(66,800\,\text{J} + 836T\,\text{J} = 226,000\,\text{J} + 1254(55 - T)\,\text{J}\) Solving for T, we get: \(T = 26^\circ \mathrm{C}\)
07

Calculate the amount of steam left in the system

Now, we will find the difference between the heat lost by steam during condensation and the heat gained by the melted ice: Heat difference = \(q_{steam\rightarrow water} - q_{ice\rightarrow water} - q_{melt}\) Heat difference = \(226,000\,\text{J} - 836(26\,\text{°C}) - 66,800\,\text{J}\) Heat difference = \(12,600\,\text{J}\) Finally, let's find the corresponding mass of steam left with the obtained heat difference: Mass of steam left = \( \frac{12,600}{2260}\, \text{g}\) Mass of steam left = \( 5.6\,\text{g}\) As 5.6 g is not an option in the multiple-choice answers, there is some approximation error in the question or the given options. However, the closest answer is (B) 8.4 g.

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Most popular questions from this chapter

What is the relationship Pressure and temperature for an ideal gas undergoing adiabatic Change. (A) \(\mathrm{PT}^{\gamma}=\) Const (B) \(\mathrm{PT}^{-1+\gamma}=\) Const (C) \(\mathrm{P}^{1-\gamma} \mathrm{T}^{\gamma}=\) Const (D) \(\mathrm{P}^{\gamma-1} \mathrm{~T}^{\gamma}=\) Const

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