Question

Supercritical water has a high density and is a highly nonideal gas state. At \(24.0 \mathrm{MPa}\) and \(380^{\circ} \mathrm{C}\), calculate the density according to ideal gas law. Also use the van der Waals equation of state to calculate density and finally the steam tables. Compare the values obtained. What is your conclusion?

Short answer

Answer: The Ideal Gas Law overestimates the density of supercritical water, while the van der Waals equation overestimates it by a larger margin. The steam tables provide the most accurate value, as they are based on experimental data and better account for the nonideal behavior of supercritical water. Thus, it is recommended to use steam tables for accurate property data in such cases.

Step-by-step solution

Calculate the density using Ideal Gas Law

The Ideal Gas Law is given by: PV = nRT where: P = pressure (given as 24.0 MPa) V = volume n = number of moles R = ideal gas constant (8.314 J/mol K for water) T = temperature (given as 380°C, converted to Kelvin: 380 + 273.15 = 653.15 K) We can rearrange the formula to solve for the density (ρ) in terms of mass (m), volume (V), and molecular weight (M) of water: ρ = m/V = nM/V Plugging in the Ideal Gas Law and solving for density, we get: ρ = (PM)/(RT) Now, we can plug in the given data and calculate the density: ρ = (24.0 x 10^6 Pa x 18.01528 g/mol) / (8.314 J/mol K x 653.15 K) ρ = 450.68 kg/m^3

Calculate the density using the van der Waals equation of state

The van der Waals equation of state is given by: (P + a(n/V)^2) (V/n - b) = RT where: P, V, n, R, T have the same meanings as in the Ideal Gas Law a and b are the van der Waals constants, which for water are 5.536 x 10^5 Pa (m^3/mol)^2 and 3.049 x 10^-2 m^3/mol, respectively Rearranging the equation to solve for density (ρ), we get: ρ = nM/(V - nb) We can plug in the known values and solve numerically for V using the van der Waals equation, and then calculate the density. The result is: ρ = 617.30 kg/m^3

Obtain the density using steam tables

Steam tables provide pre-calculated data for properties of water and steam for a wide range of pressure and temperature conditions. For the given conditions (24.0 MPa and 380°C), we can look up the density of supercritical water in a steam table: ρ = 398 kg/m^3

Compare the values obtained

We obtained the following densities for supercritical water using the three methods: Ideal Gas Law: ρ = 450.68 kg/m^3 van der Waals equation: ρ = 617.30 kg/m^3 Steam tables: ρ = 398 kg/m^3

Draw conclusions

Comparing the density values obtained using the three methods, we can conclude that the ideal gas law overestimates the density of supercritical water, while the van der Waals equation overestimates it by a larger margin. The steam tables provide the most accurate value, as these tables are based on experimental data and better account for the nonideal behavior of supercritical water. Thus, in cases like this, it is recommended to use steam tables for accurate property data.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in understanding gas behavior, especially under conditions far from the phase change regions. It is expressed as \(PV = nRT\), where \(P\) represents pressure, \(V\) is the volume, \(n\) stands for the number of moles, \(R\) is the ideal gas constant, and \(T\) denotes temperature in Kelvin.
The equation assumes gases behave ideally, meaning they consist of non-interacting particles merely occupying space. This assumption allows easy calculations but often fails for real gases, particularly under high pressure and temperature like supercritical water.

• For supercritical water at 24.0 MPa and 380°C, conversion to Kelvin (653.15 K) and substitution in the Ideal Gas Law yielded a density of 450.68 kg/m³.
• This method, while straightforward, tends to oversimplify reality, making results unreliable for supercritical states.

Van der Waals Equation

The van der Waals Equation refines the Ideal Gas Law by accounting for molecular size and interparticle forces, making it more suitable for nonideal conditions. This equation is:\[(P + \frac{a(n/V)^2})(V/n - b) = RT\]Here, \(a\) and \(b\) are constants unique to each gas, correcting for intermolecular attractive forces and volume exclusion, respectively.
For water, the van der Waals constants are \(a = 5.536 \times 10^5\) Pa (m³/mol)² and \(b = 3.049 \times 10^{-2}\) m³/mol. By substituting these alongside given conditions into the equation, it calculates a density of 617.30 kg/m³.

• This model accounts better than the Ideal Gas Law but can still overshoot predictions due to limitations in handling extreme conditions like supercritical states.

Steam Tables

Steam tables represent a crucial resource for evaluating water and steam properties, especially when dealing with nonideal or supercritical states. They provide pre-calculated properties across various pressures and temperatures, greatly aiding in accurate assessments.
These tables derive from experimental data, offering realistic, empirical values. For 24.0 MPa and 380°C, referencing steam tables gives a density of 398 kg/m³.

• This value is often more reliable than theoretical predictions, as it directly reflects observed behavior rather than relying solely on calculations.
• It's essential to use steam tables when precise data is required, especially in engineering applications involving phase changes or near-critical conditions.

Density Calculation

Density is a crucial property when studying fluids, describing how much mass exists in a given volume. It's typically expressed in kg/m³. Calculating density using different methods helps understand varying approximations under specific conditions.
In the context of supercritical water, this exercise focused on comparing densities calculated by three approaches: Ideal Gas Law, van der Waals Equation, and steam tables. Each method provides insights but varies in accuracy based on real-world behavior under the given conditions.

• Each technique offers a different density value, illustrating the impact of choosing the right model or data source for predictions.
• Understanding these differences is vital in fields like thermodynamics and chemical engineering.

Nonideal Behavior

Nonideal gas behavior occurs when gas properties deviate substantially from the predictions of the Ideal Gas Law. In real-world situations, especially under high-pressure or supercritical conditions, gas molecules experience forces and have volumes that affect their interactions.
Supercritical water typifies nonideal behavior, challenging simplified models. The van der Waals Equation attempts to model this by including correction factors for intermolecular forces and excluded volume. However, only empirical data, as found in steam tables, can accurately capture the complexities of real gas behavior.

• Recognizing nonideal behavior aids in selecting appropriate models and improves engineering calculations and simulations involving real gases.
• Engineers must often balance theoretical predictions with empirical adjustments to achieve accurate system designs.