Question

At what depth will ascending mantle rock with a temperature of \(1600 \mathrm{~K}\) melt if the equation for the solidus temperature \(T\) is $$ T(K)=1500+0.12 p(\mathrm{MPa}) $$ Assume \(\rho=3300 \mathrm{~kg} \mathrm{~m}^{-3}, g=10 \mathrm{~m} \mathrm{~s}^{-2}\), and the man- tle rock ascends at constant temperature.

Short answer

The ascending mantle rock will melt at a depth of approximately 25239.5 meters.

Step-by-step solution

Understand the Solidus Equation

The solidus equation given is \( T(K) = 1500 + 0.12p(\mathrm{MPa}) \). This describes the temperature at which the rock begins to melt as a function of pressure \( p \), measured in megapascals (MPa). We need to find the pressure at which the rock's temperature (1600 K) equals the solidus temperature.

Set Up the Equation

Set the solidus temperature equal to the ascending rock temperature: \( 1600 = 1500 + 0.12p \). Solve this equation for \( p \) to find the pressure at which the rock will melt.

Solve for Pressure \( p \)

Subtract 1500 from both sides: \( 1600 - 1500 = 0.12p \), which simplifies to \( 100 = 0.12p \). Solve for \( p \) by dividing both sides by 0.12: \( p = \frac{100}{0.12} = 833.33 \mathrm{MPa} \).

Relate Pressure to Depth

Pressure \( p \) is related to depth \( d \) by the equation \( p = \rho g d \), where \( \rho \) is the density of the mantle rock (3300 kg/m³) and \( g \) is the acceleration due to gravity (10 m/s²). Plug in \( p = 833.33 \times 10^6 \mathrm{Pa} \), \( \rho = 3300 \mathrm{kg/m^3} \), and \( g = 10 \mathrm{m/s^2} \): \( 833.33 \times 10^6 = 3300 \times 10 \times d \).

Solve for Depth \( d \)

Solve the equation for \( d \): \( d = \frac{833.33 \times 10^6}{3300 \times 10} \). Calculate to find \( d \): \( d = \frac{833333000}{33000} \approx 25239.5 \mathrm{m} \). So, the depth is approximately 25239.5 meters.

Key concepts

Mantle Dynamics

Mantle dynamics is the study of the flowing movement and behavior of the Earth's mantle. This layer, situated between the Earth's crust and core, is primarily composed of silicate rocks, and despite being solid, it behaves plastically over geological timescales. This plastic behavior is due to the high temperatures and pressures within the mantle, allowing the rocks to deform slowly and influence surface geologic activity. As the mantle moves due to convection currents driven by heat from the core, it plays a key role in plate tectonics, continental drift, and volcanic activity.
Understanding mantle dynamics is essential for geophysicists as it helps explain the processes that form mountains, ocean basins, and earthquakes. Additionally, mantle flow helps redistribute Earth's internal heat, impacting its magnetic field and surface conditions.

Solidus Temperature

The solidus temperature is the lowest temperature at which a solid material begins to melt. In geological terms, it specifically refers to the temperature at which mantle rocks start to melt as they ascend closer to the Earth's surface. As rocks rise, the pressure decreases and the solidus temperature becomes critical to understanding when the rock will begin to transition from solid to liquid.
The concept of solidus temperature is vital for understanding mantle melting and magma generation. In our exercise, the solidus equation is expressed as \( T(K) = 1500 + 0.12p(\mathrm{MPa}) \), indicating how pressure influences the exact temperature at which melting begins. This equation tells us that higher pressures increase the temperature required for the mantle to start melting. Thus, understanding the solidus temperature helps predict volcanic activity and the formation of new crustal material.

Pressure-Depth Relationship

In geophysics, the pressure-depth relationship helps determine how pressure varies with depth in the Earth. This is important because Earth's pressure dictates many geological processes, including rock melting. Pressure increases as you go deeper into the Earth, primarily due to the weight of the overlying materials.
In our problem, pressure is calculated using the equation \( p = \rho g d \), where \( \rho \) is the density (3300 kg/m³), \( g \) is the gravitational acceleration (10 m/s²), and \( d \) is the depth. By rearranging this formula, we can solve for the depth at which specific pressures occur, helping us understand at what depths certain geological phenomena occur, such as the melting point of ascending mantle rocks as calculated in the exercise.

Melting in Geophysics

Melting in geophysics refers to the transition of solid geological materials into liquid form, primarily due to changes in temperature and pressure as you move through the Earth's layers. This process occurs when ascending mantle rocks reach their solidus temperature at specific pressures. As rocks move into regions of lower pressure, they require less heat to begin melting, which produces magma.
The formation of magma has significant implications for volcanic activity and the formation of igneous rocks, which in turn shape the Earth's surface. By studying and understanding melting processes, geophysicists can gain insights into the dynamics of the Earth's interior and its influence on surface geology. Our exercise provides a practical application of these concepts, calculating the depth at which mantle rocks begin to melt under given conditions.