A certain quantity of gas is heated at constant pressure from 35°C to 185°C. Estimate the amount of heat transferred, ideal work done, change in internal energy, when initial volume of gas is 0.6 m³.

To solve this problem, we will use the ideal gas laws and thermodynamic relations for an ideal gas heated at constant pressure. Let's break it down into three main parts:

1. **Heat Transferred (\(Q\))**:
   For an ideal gas at constant pressure, the amount of heat transferred can be calculated using:
   \[
   Q = n C_p \Delta T
   \]
   where:
   - \(n\) is the number of moles of gas,
   - \(C_p\) is the specific heat capacity at constant pressure,
   - \(\Delta T\) is the change in temperature.

   However, we need to find \(n\) and we will use the formula:
   \[
   n = \frac{PV}{RT}
   \]
   where:
   - \(P\) is the pressure,
   - \(V\) is the initial volume,
   - \(R\) is the universal gas constant \(= 8.314 \, \text{J/(mol·K)}\),
   - \(T\) is the initial temperature in Kelvin.

2. **Work Done by the Gas (\(W\))**:
   For an isobaric process (constant pressure), the work done by the gas is given by:
   \[
   W = P \Delta V
   \]
   or
   \[
   W = nR \Delta T
   \]

3. **Change in Internal Energy (\(\Delta U\))**:
   For an ideal gas, the change in internal energy is given by:
   \[
   \Delta U = n C_v \Delta T
   \]
   where \(C_v\) is the specific heat capacity at constant volume.

Since \(C_p - C_v = R\), we can use the relationship:
\[
\Delta U = Q - W
\]

Let's calculate step-by-step.

### Step 1: Convert Temperatures to Kelvin
\[
T_1 = 35^\circ C + 273.15 = 308.15 \, K
\]
\[
T_2 = 185^\circ C + 273.15 = 458.15 \, K
\]

### Step 2: Find the Number of Moles (\(n\))
To calculate \(n\), we need the pressure \(P\). Since it is not given, we will assume standard atmospheric pressure (1 atm = 101325 Pa).

\[
n = \frac{PV}{RT_1} = \frac{101325 \times 0.6}{8.314 \times 308.15}
\]

Let's compute \(n\).

### Step 3: Calculate Heat Transferred (\(Q\))
Once we have \(n\), we can calculate \(Q\) using:
\[
Q = n C_p \Delta T
\]
For a diatomic ideal gas (e.g., air), \(C_p \approx 29.1 \, \text{J/(mol·K)}\).

\[
\Delta T = T_2 - T_1 = 458.15 - 308.15 = 150 \, K
\]

### Step 4: Calculate Work Done (\(W\))
\[
W = nR \Delta T
\]

### Step 5: Calculate Change in Internal Energy (\(\Delta U\))
\[
\Delta U = Q - W
\]

Let's perform the calculations.Here are the results for the thermodynamic quantities:

1. **Number of moles (\(n\))**: \(23.73 \, \text{mol}\)
2. **Heat Transferred (\(Q\))**: \(103,581 \, \text{J}\) (or \(103.6 \, \text{kJ}\))
3. **Work Done by the Gas (\(W\))**: \(29,594 \, \text{J}\) (or \(29.6 \, \text{kJ}\))
4. **Change in Internal Energy (\(\Delta U\))**: \(73,987 \, \text{J}\) (or \(74.0 \, \text{kJ}\))

### Summary
- The heat transferred to the gas during heating is approximately \(103.6 \, \text{kJ}\).
- The ideal work done by the gas due to expansion is around \(29.6 \, \text{kJ}\).
- The change in internal energy, which accounts for both the work done and the heat added, is about \(74.0 \, \text{kJ}\).

Would you like more details or have any questions?

### Related Questions:
1. What is the relationship between \(C_p\) and \(C_v\) for an ideal gas?
2. How does the specific heat capacity affect the heat transferred to a gas?
3. What would be the impact on work done if the gas was monatomic instead of diatomic?
4. How is the first law of thermodynamics applied in this problem?
5. How can these calculations be modified for non-ideal gases?

### Tip:
Remember that for an ideal gas, the internal energy change depends only on the temperature change, not on the volume or pressure changes.

Estimate the heat transferred, work done, and change in internal energy when a gas is heated from 35°C to 185°C at constant pressure. This detailed thermodynamics problem includes step-by-step calculations using the ideal gas law and related formulas. Suitable for college-level thermodynamics or physics courses.

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