Final answer:
The verification that pressure has units of energy per unit volume is demonstrated by the multiplication of pressure's unit (N/m²) with volume's unit (m³), resulting in joules (J), the unit of energy, divided by cubic meters (m³), thus confirming pressure's units as energy per unit volume.
Step-by-step explanation:
To verify that pressure has units of energy per unit volume, we first consider that pressure is defined as force per unit area (P = F/A), with units of newtons per square meter (N/m²). When considering the volume, which has units of cubic meters (m³), we can see that the work done (W) when a volume changes under constant pressure can be expressed as the product of pressure and volume change (W = PΔV).
Using the units for pressure (N/m²) and volume (m³), we observe that multiplying them gives us units of N·m, which are equivalent to joules (J), the SI unit for energy. Therefore, when we consider pressure acting on a given volume, we get units of energy (J = N·m). Dividing both sides by volume (m³), we end up with J/m³, which shows that pressure indeed has units of energy per unit volume.
Additionally, in the context of thermodynamics, such as through the ideal gas law (PV = nRT), it is noted that the product of pressure and volume (PV) has units of energy—further reinforcing the relationship between pressure and energy density. Moreover, in fluid dynamics, Bernoulli's equation presents pressure as a form of gravitational potential energy per unit volume (pgh), cementing the concept that pressure is energetically described per unit volume.