The following lists all application pools in IIS and matches with their associated w3wp.exe process id.

ListRunningAppPools.ps1

Import-Module -Name WebAdministration

$appPoolsPath = "IIS:\AppPools"

foreach($appPool in Get-ChildItem -Path $appPoolsPath) {
    if($appPool.state -eq "Started") {
        $processes = Get-ChildItem -Path "$($appPoolsPath)\$($appPool.Name)\WorkerProcesses"
        foreach($process in $processes) {
            [pscustomobject]@{
                "AppPool" = $appPool.Name
                "ProcessId" = $process.ProcessId
                "State" = $process.State
                "StartTime" = $process.StartTime
            }
        }
    }
}