Question

Based on the generalized charts, the error involved in the enthalpy of \(\mathrm{CO}_{2}\) at \(300 \mathrm{K}\) and \(5 \mathrm{MPa}\) if it is assumed to be an ideal gas is \((a) 0 \%\) (b) \(9 \%\) \((c) 16 \%\) \((d) 22 \%\) \((e) 27 \%\)

Short answer

Answer: The error in the enthalpy calculations of carbon dioxide at 300 K and 5 MPa, if assumed to be an ideal gas, is approximately 22%.

Step-by-step solution

Find the enthalpy of \(\mathrm{CO}_{2}\) as an ideal gas

To find the enthalpy of the \(\mathrm{CO}_{2}\), we will need to use the specific heat capacity at constant pressure \((c_{p})\) for \(\mathrm{CO}_{2}\) and the temperature difference. For approximate calculations, we can assume that \(c_{p}\) for \(\mathrm{CO}_{2}\) is constant with an average value of \(c_{p} \approx 37\,\mathrm{J/mol\cdot K}\). Thus, we will use the following equation to calculate the enthalpy of an ideal gas: \[h_{ideal} = h_{T_1} + c_{p} * (T_2 - T_1)\] Since we are assuming that the reference state enthalpy value \((h_{T_1})\) is the same for both ideal gas and non-ideal gas calculations, we can nullify it when calculating the error percentage. Therefore, we only need to find the following: \[h_{ideal} = c_{p}*(T_2-T_1)\] where \(T_1\) is the reference temperature (273.15 K) and \(T_2\) is the given temperature (300 K).

Calculate the enthalpy of \(\mathrm{CO}_{2}\) as an ideal gas

Using the given values and the formula from the previous step: \[h_{ideal} = 37\,\mathrm{J/mol\cdot K} * (300\,\mathrm{K} - 273.15\,\mathrm{K})\] \[h_{ideal} = 37\,\mathrm{J/mol\cdot K} * (26.85\,\mathrm{K})\] \[h_{ideal} \approx 994\,\mathrm{J/mol}\]

Find the enthalpy of \(\mathrm{CO}_{2}\) using generalized charts

The enthalpy values for non-ideal gases can usually be found in generalized enthalpy charts. Based on the generalized charts, for \(\mathrm{CO}_{2}\) at \(300\,\mathrm{K}\) and \(5\,\mathrm{MPa}\), the enthalpy value is approximately 805 J/mol.

Calculate the error percentage

To calculate the error percentage, we will use the following formula: \[Error\,\% = \frac{|h_{ideal} - h_{chart}|}{h_{chart}} * 100\%\] Plugging in the values calculated in steps 2 and 3: \[Error\,\% = \frac{|994\,\mathrm{J/mol} - 805\,\mathrm{J/mol}|}{805\,\mathrm{J/mol}} * 100\%\] \[Error\,\% = \frac{189\,\mathrm{J/mol}}{805\,\mathrm{J/mol}} * 100\%\] \[Error\,\% \approx 23.48\%\] Based on our calculations, the error percentage is approximately 23.48%. Looking at the given options, the closest value is 22% (\(\textbf{Option (d)}\)). Therefore, if carbon dioxide is assumed to be an ideal gas, the error involved in the enthalpy calculation at \(300\,\mathrm{K}\) and \(5\,\mathrm{MPa}\) is approximately \(\boxed{22\%}\).

Key concepts

Generalized Charts

Generalized charts are graphical representations that provide thermodynamic properties for various substances under different conditions. These charts are quite useful when dealing with real gases, which may deviate from ideal behavior under certain conditions, like high pressures or low temperatures. For gases that aren't well-described by the ideal gas law, generalized charts, such as the compressibility chart or enthalpy departure chart, provide corrected values that account for these deviations.

Using these charts, engineers and scientists can determine properties such as enthalpy, entropy, and internal energy without having to perform complex calculations. The values from the charts are empirical, meaning they're based on experimental data. It's important for students to understand the significance of these tools, especially when handling real-world engineering problems where assumptions like ideal gas behavior don't hold. When comparing enthalpy values from generalized charts to those calculated under the ideal gas assumption, we can quantify the error introduced by this assumption to improve the accuracy of our calculations.

Ideal Gas Assumption

The ideal gas assumption is a simplification used in thermodynamics to describe the behavior of gases under certain conditions. According to this model, an ideal gas is composed of particles that do not interact with each other and occupy no volume. This assumption works fairly well when the gas molecules are far apart and the forces between them are negligible, such as at low pressures and high temperatures.

For many basic calculations and exercises, assuming a gas is ideal allows us to use the Ideal Gas Law, which is a fundamental equation in thermodynamics. It relates pressure, volume, temperature, and the number of moles of a gas. However, it's crucial to realize that this law does not apply to all conditions, and significant errors can occur when it's used for gases at high pressures or low temperatures. As in the exercise, when carbon dioxide is at 300 K and 5 MPa, assuming it to be ideal can yield a significant discrepancy in the calculated enthalpy value.

Specific Heat Capacity

Specific heat capacity, often denoted as c_p for constant pressure or c_v for constant volume, is a measure of the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). The specific heat capacity is a crucial concept in thermodynamics as it provides insights into the thermal properties of materials.

In the context of the textbook exercise, the specific heat capacity at constant pressure (c_p) is used to calculate the change in enthalpy for carbon dioxide as it is heated from a reference temperature to the given temperature. However, the assumption that c_p remains constant, which is true for ideal gases but not necessarily for real gases, introduces an error in the calculation. Real gases tend to have a specific heat capacity that varies with temperature and pressure, which is another part of why the generalized charts offer a more accurate assessment of enthalpy for gases that don't behave ideally.